This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we have taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain from automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attribution The Google "watermark" you see on each file is essential for informing people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liability can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at |http : //books . google . com/
I'^m^^^imi
%^<
mt
^»^ila»-^ ^^^
i|:
^^f
jdfc[^jfc "^^ •^^ •^^ •^^3®* •^w i^r •*«'»■
Hj^^^. -|j* -^» -§► -J: -§• - ^ - !^ ' ■ '?•
■ • :#:
.5- .ffr.
ea BO • A
r / LOGARITHMOLOGIAr^7
ORTHEWHOLE , M 3> 8
i DO C T R I N E,
[> xo Car I t h m s^
ft.
Common and hogiftical^
IN
THE.ORY and PRACTICE.
IN THREE PARTS.
Part. I. The THEORY of Logarithms;
Shewing their Nature, Origin, Cotfftru^lion, and Properties,
demonftrated in varioas Methods, was. i. By Plain Arithme-
tic. 2. By the Logarithmic Carve. 3. By Dr. Halley's
Infinite Scries. 4. By Fluxions, c. By ^c Properties of the
Hyperbola. 6. fey the Equiangular Spirnl. 7. By a Loga-
rithmic infpeftional Scale of twenty- two Inches length. With
the Conftru^ion of the artificial Lines of Numbers, Sines,
and Tangents. Alfo the Nature and Conftrudlion of Logiftical
Logarithms. The whole illutlrated and made eafy by many
<and fuitable ExaB^ples.
Part II. The PRAXIS of Logarithms;
Wherein all the Rules and Operations of Logarithmical Arith-
metic, both Common and Logittical, by Numbers and Inftru-
ments, are copioufly exemplified. Together with the Ap-
plication thereof to the feveral Branches of Mathematical
Learning.
PartIIL a Three-fold canon of Logarithms;
In a new and more compendious Method than any cxtint ;
r I. A Canon of Logarithms of Natural Numbers.
* Vi%. X 2. A Canon of Logarithms of S 1 n fis and Tangents.
\ ^3. A Table of Logistical Logarithms.
The whole being a CompUat Syftem of this mod ufeful Art ;
and enrich'd with all the Improvements therein from its Ori-
ginal to the Prefcnt Time.
By B E N 7 A M I 1^ MARTIN,
Author of the Philological Library of Literary Arts and
Sciences^ &c.
LONDON:
Print^ for J. Hodges, at the hooking-Qlafs on London-Bridge*
M.pcg.xxxx.
.J5
■\\
v1
Ul
O
^'3 'S3
^f
m':WP.
^ THE
1
iP R E F A G E.
TBO Logarithms fnay juftfy be efteenid
the principal Invention ^modern Ages,
on account of their excellent and tnojl
\[ fcxtcniivc Ufe in Mathematical Literature, yet
it may ^ith equal Truth be faid^ that little
inoh is known of them^ generally Jpeaking^
than theif praftical Ufe infome Rules ^/com-
mon Arithmetic and trigonometrical Calcula-
tions i {and how few are perfe£i in this !)
For (faith the great Improver of this Art^ the
^ learned T)r. Halley) " / fivU very few of
*^ thofe^ whomakeconftantufeofLo^^mhms^
" have attain d an adequate Notion of them i
^^ to know how to make or examine them^ of
" to underfland the Extent ^/ the \5i^ of
" them : contenting themfelves with the Ta-
*' bles of them, as they find them, without
^^ daring to queftion them, or caring to know
'^ how to redify themy Jbould they be found
" amifs i beingy I fuppofe,^ under the Appre-
'^ herfion of fame great ^:jficultjf therein, &c/'
A 2 ^ For
iv PREFACE.
For the fake of fuch Terfons the following
Trcatife is principally intended i wherein they
fwill find every thing necejfary relating Jo the
Yheory and VxoiQticc of this admirable Art.
For, as to the Theory, {the principal Part,
and fo very rarely known) I have exhibited
all the Methcxls, whereby it has been taught
and explain d by the Inventor, and fever al
Improvers thereof fince his time y as by the
Extraftion of Roots i by the Logarithmic Curve
from jDr. Kcil > by an Infinite Scnts^from T>r.
Halky i by the Method of Iluxions, from Mr.
Ditton i by the Hyperbola, /r^w? Mr. Domky i
by the Equiangular Spiral, from Halley, Wallis,
h'C. by the /^rg*^ Logarithmic Scale, of my awn
conftruding : the like to which, for Large-
nefs, was never before publijhed j for a com-
pleat Account of which fee Chap. X. of the
Theory.
I fay, by all thefe various Methods / have
endeavour d to explain, illuftrate, and facili-
tate the Knowledge of the Nature, Proper-
tics, and Conftrudion of thofe excellent Num-
bers, called Logarithms. / have alfo exem-
plified the Manner of making Logarithms yj?r
the prime Numbers, by many and different
Examples, afid in feveral ways s and have
taken all poffible Care to render this mojl ab-
ftrufe a7id difficult ^art, as eafy and intelli- •
gibic, as the Nature of the SubjeSt will
admit.
Having
PREFACE. V
Having thus explain d the Nature of Lo-
garithms, / then fiew how they are laid wn
Inftruments, and thereby the Cooftruftion of
the Artificial Lines of Numbers, Sines, and
Tangents i firji contrived by the famous Geo-
meter y Mr. Gunter of Grefliam-Collegc s an4
for that reafon they are ftill called Guntcr's
Line, and all together Garner's Scale.
Lajilyy I have largely explain' d the N^
ture^ and Jhewn the Manner of making or
conJiru£ting the tx)giftical Logarithms, accor-
ding to ShakerleyV and Stictt's Form thererf^
and which I have notfeen done by any otier
Hand.
Thefe things together willy I hopCy be al^
low'd to make a regular, univerfal, and com-
pleat Theory of Logarithms, common and lo-
giftical, for Integers and FraiiionSy numeri-
cal and inftrumental ; and fuchy as for Bre-
vity, yet Copioufnefs and Variety, has not
been before extant.
As touching the Praxis, or Ufe ^Loga-
rithms, which makes the fecond Tart of this
PForky I have made it as compleat and per-
fe£t as poffible^ having illuftrated all the
Rules of Praftice with all theV^ncxy ofUx-
ampJcs / could devifey that were necejfary.
■^nd that none may be unapprised of the mofi
^xtcxifivc Service of Logarithms in the Mathe-
^^^tical Difdplines, I have applied them to the
Arithmetic
in t^REFACE.
Arithmetic of all kinds of Number i ^ to Tri"
gonomctry, in the Solution df all Cafes of
Plain and Spherical Triangles i to Mcrc^tors
Sailing particularly y jhowinghow all its Cafes
may be refolvd folely by the Canon of Loga-
rithmic Tangents ; to the Mcnfuration 4if Su-
perficies i^«/Solids^ ^c: All which are fun-
damental Operations, and may each of them
be extended or branched out into particular
Sciences i but that would have been too te-
dious a Taskj and not ahfolutely necejfary to
^ ^ejign. I have there fare only applied the
Doftrine of Mcnluration to the Arts of Gzn^-
ing, Timber-Meafure, ^«// Surveying i becauje
they are the mo(l common and n^ceffary Arts
in Life s and beeaufe the Ufe of Gunter'S
Scale, and Slidiag-Rule {though before fully
taught^ and all along applied i yet) in theni
is more extenfive and various than in any o-
ther Arts: and therefore I have taken care
not only tojkew all the different ways ofufing
thofe inftruments, but likewife the Rationale:
of every Operation ^ a Matter of the greatefl
Importance^ and too often negle^ed^ in Books
which treat thereof
Laftly, I have in the laft Chapter given a
Variety of Examples of the life of Logiftical
IvOgarithms in ?^^ practical Parts i?/*Aftronomy;
4)oih with refpeB to Time and Motion, have
made it appear that Street's Logiftical Loga-
rithms arifwer all the Ends of Slaakerlcy j i
^nd how they are * to be ufed along with the
Common Logarithms of Numbers, Sines, and
Tangents;
PREFACE. vii
Tangents. And throughout this fecoAd Part,
as w^ll as tj^e firft, you will fnd 4 great
Variety of new and ufeful Particulars^ not
here tfi be exprefsd. ♦
The Third Part of this Work canfijls ^f a
xjtixt^'iold Canon of Logarithms, viz. (i.) Of
fomnum Numbers from i to loooo/ oAd is
Gifficient for any Number under 1 0000000^
proper Rules. (2). Of Sines 49?^ Tangents /v
every Degree and Minute of the Quadrant.
(i-) ^f L^i^i^'i Logarithms of Mr. StrcctV
Form. Concerning which Tables y Ifballcnly
Qbfsrve two things in general^ viz,
Firfly that they are here contrived in a
new and moji compendious Form, equally cafy
and ufcful as thofe of the common Formi thd
in this they take up but one half the room^ as
they do in that. An Abbreviation very conh
modiouSy and I hope will prove acceptable.
Secondly ^ tiie Cwrte^nefs of thefe is a
matter of the tafl Concerny and the great eft
Argument to recommend them. In order to
prove thiSy J need only fay^ that thofe large
Tables of Mr. Sherwin j ape granted to be the
moft correSi of any extant y from the maft
(^tQfal^and exquiiite Method he took to make
ihemfoy which fee in his Preface.
From thefe large Tables thus correify I
have made minCy every Figure of the two
^rft Tables with my own Hands in doing
which
vifi PREFACE.
whichy I difcoverd fever al Error s, here and
there y as I went along, in them^ as exuEt as
they were, which accordingly correcting in
mine, lean, I prefuthe, juftly pronounce my
Tables thi mofi certain and exaft, as well as
the moft compendious of any in being.
Having thus largely declared the fever al
Tarts of this Work, and Jhewn the Reader
whdt an ufeful Variety he may expert to meet
with both in the Theory, Praxis, and Tables,
of this moft 4ifeful and excellent Art } I muft
leave it to himfelf to ufe or reject it, as he
(ball judge of the Merits thereof
THE
(«)
THE
TABLE 6f CONTENTS.
PART L
CHAP. I.
Op /i6^ Definition, Origin, and Natvvle
^Logarithms. p. i
CHAP. IL
Of the Method cf making a Table of Loga-
rithms i^ Plain Arithmetic. 9
C H A P. III.
T'be Doctrine ofibe Nature and Prope rties
of luOGAK I THUS farther explained and illuftratcd
by means of tbe Logarithmic Curve. 18
C H A P.^ IV.
2*i&^NATURE (?/■ Logarithms and their IitvtCES^
when tbe Numbers are Fractions, farther ex^
plained by tbe hoGAKiTHMic Curve. 25
C H A P. V.
fTbe Original Conftruftion of Logarithms by tbe
Lord Ne per, and tbe Change thereof to tbe pre-
fent Form by Himfelf tf»i Mr. Briggs, explained
and illuftratcd by tbe Looa rithmicCwrve. 35
CHAP. VL
yf Method of Conftrufting the Logarithms, de-
, rived and demonfirated from tbe Nature of Num-
^ B^RS only^ by Dr. Edmund Halle y. 40
a CHAP.
X CONTENTS,,.
CHAP. VII.
The JLoc arithmicSeries aforegoing demonfirated
alfo by Fluxjons^; and from the Nature of the
HypERpoLA. * p. 50
CHAP. VIII.
?J^ Method of Conftruifing Log a Kir hub by the
Infjnite Series, exemplified and illufirated. 53
C H A P. IX.
Qf /^^* Logarithmic Spiral ; and the Nature
and Qor[&xw&\on of a Table of Meridional
Parts or Nauticfl Meridjan Line, deduced
therefrom. 64
CHAP. X.
Of the Conftruiftion of a Large Logarithmic
Sc ALE, exhibiting^ by Infpeftion, a diftind Idea
^/i>^ Nature ^» J Agreement of Numbers and
their LoQARiTHMS. 71
CHAP. XI.
Of the Conftnwftion of the Artificial Lines ^N^m-
b|:rs, SiNfs, ^;/i Tangents, ^ i»^tf»i ^/Lor
GARITHMS. '73
CHAP. XII.
Of the Manner of ufing the Tables of Log a-?
' RiTHMs/» Practice » and of the Pre requi-
sites /i^^r^/^. 835
CHAP. ?crii
Of the Origin and Conftrudion of Shake rleyV
>?»^St;ieetVLogisticalI^ooarithm3. 93
!> A R T IL
G H A P. I.
<y /i&? Rules u/ Addition, SuBSTRACTioii,Mut,"r
. TjPLicATjoN, ^nd pivisioN of the Indices ^»
LoGARItHMS, * 103
C H A F.
r
CONTENTS. si
CHAP. II.
f^MutTIPLICATlON tf)iiDlT|SION ^ Wh6Ls
NuMBEI^S^»^DECIM4I-SbyL0OARITHMS.I0$
CHAP. III.
Of Raifing Powers andtU £xt b act ion ^Rqot$
>y LOGARITHMS. P-II5
CHAP. IV.
Ofthi various Ru lbs ^t/Troport k>k, tfni of findings
M£A9PROPORTIONA|.si^LOGARITHM8.I2C
CHAP. V.
ISiMPLB Interest by Logarithms. 128
CHAP. VL
Compound Interest by Logarithms. 138
CHAP. VIL
ViTLG A R Fractions i^ Logarithms 148
CHAP. VIII.
*Puo&£crMAL Akithu^t ic pirfotn^d iy Looa«
RITHMS. 152
CHAP. IX.
flT^^OpERATioN ^/itf common Rules ^ Arith-
metic ty Inst RujfE NTS j viz. the Loga-
rithmic Scale, tf»rfGuNTER's Line mtb
/i&^ Compasses, ^»^^ir/i&^SLiDiNG-RuLE. 158
C H A P. X.
^keiJbeth the Analogies for the Solution of all
the Cases of Plain and Spherical Tri an-p
G L E s, both Right and Oblique angled. 1 67
CHAR XI.
y'ifrtf Solution of Plain TRiANGLE&iytbeCA^
!f O N (?/ Lo q 4 R I T H M I C S I N E S ^l»^ T A N G ¥ N T S ;
^GuNTEB*sScALE^WCOMPASSES,tf»^^JF/i&^
l idinc-Rule. 173
C H A p. XII.
^e$OLvxiQN of Spherical Triangles ly
Logarithms, iy Gunter's Scai;-e, and by
/i>^ Sliding-Rule. 1*8$
" ' ^ H A P.
I
301 CONTENTS.
CHAP. XIII. :
M?RC A tor's Sailing performed ly ib^ QakoiI
£/■ Logarithmic Tangents^ without the
Meridional Parts. p. 2ocr
CHAP.. Xi V.
Of the Mensuration (j/'SuperfIcies and So-
L I D Bo D I E s ^j^ Logarithms. 207
C H A P. XV. .
^eDo&rine ^Mensuration applied to Gaug-
ing, Meafuring Timber, and Surveying;
wherein /i&^ Praftical Ufe of. theVhAiti Scale» *
and Sliding Ru l e for tbefe Purpofes^ is clearly
^xplain^d. 222
CHAP. XVI.
^n>e Prafticaf Usfi of the Logistical Loga-
rithms. 239
-ii^
f. PART ill.
yf Canon yLoGARiTHMs/?r Natural i^uM^
BERs, /||ij» I /^ loooo. 249
A CanonT?/' Logarithmic Sines ind Tan-
gents. ^
A Canon or Table of Logistical Loga-
rithms.
L O G A-
]
i
LOGARITHMOLOGIA.
PART I.
The 7%eory of Log AKiTHMs*
C H A P. I.
Of the Definitioriy Origin^ and Natun of
Logarithms. •
I "T
^HE bcft Definitim of thofe Numbem
we caii Logarithms^ is containM in thb
very Name or JVerd {Logariibm) it*
felf 5 for it is compofed of die two
Greek Word« xiym «piO|uo\, which properly or lite-
rally %nify, a Number of Rations: and a Logarithm
is no other than a Number, which denotes or fliews
what Number of Rations is contained between Unity^
and fhat^ Number of which it is ikid to be the Lo^
garitbm.
2. Whence 'tis evident^ that In order to have a
clear Notion of Logarithms^ 'tis abfolutelv neceffiuy
to underftand firft, and that very well, what is
meant by the Word Ratic^ ot R^atio^s^ as here ufed
in the Definition of Logarithms, and making an ef«
fcntial part thereof.
3. Ratioy then^ is a certain mutual Habitude of
Magnitude^ of the fame kind^ according to ^antity.
This is Euclid^s Definition : in which four things
muft be obferv'd j as (i.) he {kith Ratio is a certain
mutuiU Habitude} by which h^ pieana no more than
B ' what
a Of the Definition, Origin^ and
whaCt we commonly call xht Proportion of any two
things of a like fort tojacb other ^ when by ys they*
arc compared together. (2.) He ufeth the general
Word Magnitude to denote, that all SubjeSfs of ^an-
tity^ 9S Numbers^ Lines^ Superfictes, and Solids^ are
capable of fuch Ratioj Habitude^ or Proportion^ as
aforefaid. (3.) He adds this Reftridion, of the fame
kind 5 thereby infinuating there can be no Ratio or
Proportion of ^antity between Magnitudes ctf*a dif-
ferent kind ; thus we cannot compare a Line to a
Superficies ; becaufe the Quantity of a Line is cfti-
mated in Length only, but the ^antity of a Super-
ficies arifeth from the Joint Confideration of Length
and Breadth, or the Produft of each, and fo im*
porteth Space ; which is entirely different from a
Line^ and therefore thefe two things cannot be the
Terms of Ratio or Comparifon. (4.) Laftly, he
fays, this Ratio is according to ^antity ; that is,
we compare Magnitudes in this Cafe, only to ob-
lerve and maintain the Proportion of Greatnefs^ or
Bulk which is between them ; or to find how often^
or how many times^ one leffer Magnitude is contained
in another greater Magnitude \ negleAing all other
Confiderations and AfFedions of the faid Magni-
tudes.
. 4. Having thus confider'd the general Nature of
Rations or Proportions ; I fhall apply it to Numbers^
as they are immediately the Subject of Logarithms.
The Ratio therefore, or Proportion of a Number to
a Number is two-fold ; for firft the Ratio of a greater
Number to a leffer may confifl in tht Addition of
fome certain Number to that lefjer Number '^ thus the
Ratio of 6 to 2 is made by adding 4 to 2. And
if from Unity you begin the conftant Addition of
the fame Number, fuppofe 2, you will then have a
3eries of Numbers, whofe Differences will be the
feme, as i, 2, 4, 6, 8, 10, 12, i^c, and fuch Num-
bers are (aid to be in Arithmetical Proportion or. Pro-
• . grejftoni
Chap. I. feature g/*LoGARiTH M5. 3
• gr^ffion 5 and this common Differenced the Terms,
as here 2 is called the Ratio of the Progreffion.
5. Secondly, the Ratio of a greater Number to a
lejfer may'confift in a Multiplication o( the Lejfer
Number by fome other Number 5 thus the Ratio of
12 to 4, is made by multiplying 4 by 3 ; and if
from Unity you begin a conftant Multiplication by
th^ fame Number^ fuppofe 2, you will then have a
Series of Numbers, as 1, 2, 4, 8, 16, 32, 64, £s?r,
which are faid to be in Geometrical Proportion^ or
Progrejfton ; and the common Multiplier^ as here 2,
is called the Geometrical Ratio of this Progreffion.
6. Wherefore in the two Series or Progreffions of
Numbers, viz.
Arith. I, 2, 4, 6, 8, 10, 12, 14, (Sc.
Geom. I, 2, 4, 8, 16, 32, 64, 128, 6?^.
*tis eafy to obferve, that as the /econd Term ex-
ceeds the firft by one Ratio, fo the third Term ex-
ceeds the firft by two Ratio's, the fourth by three
Ratio's, the fifth by four Ratio's; fe? f . Thus in the
Arithmetical Series, the Ratio of 4 to i, is dottle
of the Ratio of 2 to i ; the Ratio of 6 to i, is /r/-
ple the Ratio of 2 to i ; the Ratio of 8 to i, is
quadruple the Ratio of 2 to i, fcfr. And in the
Geometrical ISeries, the Ratio of 4 to i is the du^.
plicate of the Ratio of 2 to i ; the Ratio of 8 to i
is triplicate of the Ratio of 2 to i, and the Ratio
of 16 to I is quadruplicate of the Ratio of 2 to i,
and fo on. Where 'tis to be obferved, that tho
Words, double^ triple^ quadruple^ &c. are proper to
the Ratio's of the Arithmetical Series^ and imply the
Addition of them only ; but the Terms duplicate^ .
triplicate^ quadruplicate^ Stc. are proper to the Geo-,
metrical Series^ and imply ,ithe Multiplication of thofc
Ratio's.
7. What has been thus far related of the Doftrine
of Ratio's, is fufiicient for our prefcnt purpofe,. vi%.
the underftanding the Nature of logarithms.: For
B 2 fuppofe
4 Of the Definition^ Origin^ and
fuppofe a Series of Numbers in Arithmetical Progref"
fton^ beginning frpm o, and whofe Ratio^ or com-
iiion Difference > is Unity or i ; and to them be
adapted a Series in Geometrical Progrejfion^ beginning
from Unity ; and whofe common Ratio be any af-
fign'd Number, fuppofe 2, as before ; then will
thofe two Series ftand as below :
. iArith.o. 1.2.3. 4- 5- 6. 7, 8. 9 &?f.
• *Geom;i. 2.4.8. 16.52. 64.128. 256.512.6?^.
8. 'Tis evident the Numbers in the firft Series
fliew the Number of Ratio's between their corre-
fpondent Numbers and Unity in the fecond Series.
For inftance, the Figure 2 in the firft Series, fliews
the Ratio's hptween its correfponding Number 4 and
I, in the lower Series, are 2 ; the Numbers 5, 7, 9,
in- the upper Series, fliew the Number of Ratio's be-
tw(*eQ their correfponding Numbers 32, 128, 512,
and I or Unity, in the lower Series, to be refpeftively
5, 7, and 9 ; or that the Ratio is fo often repeated
from Unity to thofe Numbers, and confequently fo
often compounded in them ; or farther, that the
Ratio of 32 to I is compounded of five times the Ra*
tio of 2 to I ; and the Ratio of 128 to i, of feven
times the Ratio of 2 to i ; and the Ratio of 512
to I, of nines times the Ratio of 2 to i.
9^ Wherefore fince the Numbers in the uj^^ Se-
ries (Hew the Number of Ratio's contain'd between
their correfponding Numbers and Unity in the lower
Series ; therefore thofe Numbers in the upper Series
in Arithmetical Ptogreflion fliall be the Lagarittms
of the Numbers in the lower Series of Geometrical
Progreffion, and that according to the Definition of
J^ogarithms beforegoing.
iQ. From the &id Series, 'tis fiuther manifeft,
that the Froduft of any two Terms in the lower
Series corrcfponds to the Sum of their rcfpeftive
Terms in the upper. Series. See the folbwing £x*
ampler
^ Arith.
Chap. I. iVij/«r^ ^Logarithms. 5
Logar. {2 + 3= 5i2+ 4== 6^3+ 6« 9.
N:Nl;m'{^''^ = 3254x i6«64-, 8 x 64=512.
Alfo if any two Numbers in the lower Series be di-
vided the one by the other, the Quotient thence a-
rifing (hall correspond to the Difference of the respec-
tive Numbers in the upper Series. Examples in
Arith. 5— 3 = 2j 6 — 2= 4; 9—3==: 6.
Geom. 324- 8s=s4i 64-^4= 165 512 -7- 8«8 64«
1 1. And univerfally, if any four contiguous Num*
bers be taken in the fecond Series, as the ProduA
of the Extremes is equal to the ProduA of the
Means ; fo in the firft Series of the four correfpon*
ding Numbers, the Sum of the Extremes will be
equal to the Sum of the Means; as in the Examples
bdow.
Arith, 1+ 4=2+3= 5; 3+ 6=s 4+ 5=5 ^.
Geom. 2 X 16=4 x 8=32 •, 8 x 64=16 x 32=512.
and vice i^erfa.
12. Again the Square^ Cube^ (^c. of any Number
in the lower Series of Geometricals will be anfwer*d
by doubkj triple^ &c. the correfponding Number
in the upper Series in Arithmetical Ratio ; for Ex-
ample ;
Square j^"*- ^'^"^ f* ^ 3= «><^^^=^-
^ *Geom, 4x4=i6i 16x16=256-, IX isai.
The Cube \ Arithmet. i x 3=3 ; 3x3= 9.
■'^^'""^tGeomet. 2x2x2=8. 8x8x8 = 512.
And the Converfe of this Article is alfo true ; as is
evident enough without example.
: Jfc^i fr appears then by thele fix laft Articles, that
jL^arifims are a Series of Numbers in Arithmetical
Pfagreffion^ fo fitted and adapted to another Series of
Numbers in Geometrical Progrejfion^ as that each
I Term
6 Of the Definition y Origin ^ and
Term of the firft Jhall expound (or be the Exponent
of) the Ratio of its correfpondent Term to Unity in
the fecond Series. And that on this very Principle :
For every Addition^ Subjiraciion^ Multiplication^ or
Divijion of the Logarithmtc Numbers there corre-
fponds a mutual Multiplication^ Divifion^ Involution^
and Exfrailion of the refpeSlive Terms in the Geo-
metrical Series.
14. Now 'tis a Matter entirely arbitrary or indif-
ferent, what Number be made tht firft Term in ei-.
ther Series; for fince tht firft are made by equal
Additions^ the latter by equal Multiplication^ be the
Ratio what it will, th^ former will ftill be the Loga-
ritbms of the latter ; as is evident in the Table ad-
joined.
Series of Logarithms.
6
I
. ' I
3.
6
7
18
2
2
5
7
12 I
10
54
4
3
7
8
17 2
20
162
8
4
9
9
22 3
30
486
16
5
II
10
27 4
40
1458
32
6
13
II
32 5
50
4374
64
7
15
12
37 6-
60
1 3 122
128
8
17
13
42 7
70
39366
256
; 9
19
14
47 8
80
I I 8098
512
10
21
15
52 9
90
354294
1024
II
23
16
57 10
100
15. Wherefore to tht fame Series of Proportion
nals there may be an infinite' Number of 5m^j or
5rtf/^J of Logarithms contrived ; and vice verfa. Yet •
of tf//thofe infinite kinds of Logarithms, only thofe.
whofe firft Term is o, and the common Difference i,
10, 100, 6fr. are adapted for ule. Becaufe if the
firft Term be a fignificant Figure, we muft necefla-
rily have refpcdl: to it in ufe 5 and fo, in this cafe,
four
Chap. I. Nature of Log AKiTU his. 7
four Terms of the Losaritbmical Series becomes un»
avoidable •, whereas if o be the firft Term, throe
other Terms only fuffice in the Multiplication of any
two Proportionals whatever: for the Sum of their
Logarithms will point out the ProduSl^ and /hew
its Place^ that is, its Difiance from, or Ratiif to,
the firft Term of that Scale of Proportionals. But
if the firft Term in the ScaljB of Logarithms were
fignificant, it muft be fubdufted from the Sum of
the Mean, in order to find the Produdl of the two
Proportionals^ 6?r. as before: and fo we (hould per-
petually have double Labour in every Operation. AH
this is evident from the difiFerent Series of Loga-
rithms in the foregoing Table.
16. This being the Nature, and fiich the won-
derful Properties of thofe Numbers called Loga-
rithms, 'tis natural to fuppofe that he who firft dif-
cover'd them, would make fuch a noble Difcovery
as fubfcrvient as poffible to the Ufes of Life, for the
general Benefit of Mankind, but more immediately of
Artifts or Mathematicians.
17. Now to do this, 'twas
necefiary to calculate and fit
a Scale or Canon of Loga-
rithms to all Numbers which
Men commonly make ufe of
in Bufinefs, that fo the Fa-
tigue and Labour of Multi-
plying^ Dividing^ &c. large
Sums or Numbers might be
avoided by only the Addition or Subftradlon, 6? r,
of Logarithms. And of confequence nothing lefs
than a general Table of Logarithms for all Numbers
betWeen Unity, or i and 1 000000 or 1 0000000, could
fuffice ; becaufe, no Number muft here be wanting,
as in the other intermitting Series i, 2,4, 8, 16, £5?^,
fince general Ufe requires them all, and therefore
their Logarithms.
18.
Proport. 1 Logarith. |
I
0.0000000
10
1. 0000000
100
2.0000000
1000
3.0000000
1 0000
4.0000000
I 00000
5.0000000
loooodo
6.0000000
8 Of the Definition^ Origin y &c. .
1 8. Towards this 'twas eafy to find Logarithms
for a Series of Numbers proceeding in a Decuple
Propmitm or i?^//^ from Unity, as i, lo, lOO, 1060,
6fr, in the Table above. For putting o r= Loga-
rithm of I 5 and I = Logarithm of ip ; the Loga-
rithm of 100 will be 2 ; and of 1000, 3 ; and fo on
from what I have already faid.
19. But the Difficulty dl confifted in finding the
Logarithms for the intermediate Numbers between
I and 10, to and 100, 100 and 1000, tfc. or thofe
of them which are called Prime Numbers^ becaufe
they being once found, the Logarithms of all others
are cafily obtained, as we Ihall fee by and by.
20. Now tho* the Numbers i, 2, 3, 4, 5, 6, 7,
8, 9, are not in Geometrical Proportion (but indeed
in an Arithmetical one) and fo their Logarithms not
to be obtained like the Logarithms of thofe which
are fo, yet as the Ratio of 10 to i, (hews only what
Diftance^ or bow far 10 is from Unity in the Scale
of Proportion, and that the Numbers 2, 3, 4, 5,
•6?f. poflefs feverally a certain part of that Diftance,
if the iaid Ratio or Diftance of lo to i, be fuppofed
;to he divided into a vaft number of equal Parts^
fiippoie looooooo, ^c. 'tis evident a certain number
lof thofe equal Parts are to be allotted to the Num-
bers 2, g^ 4, 5, 6?r. which (hall as truly exprefs the
J}^ja$ces from, or the Rations of thofe feveral Num-
bws to Unity, as that of 10 to Unity is exprefs'd by
^ wiiole Number 1 0000000.
^i. This indefinite Number (loooooo, (Sc) of
ijpial Part 5^ into which the whole Ratio of 10 to i,
fe divided, may be conceived as a Number of fo
tnany finall Rations or RatiuncuU^ fince they are all
equal to each other. And fo fince the whole Num-
ber of RatiuncuU between 10 and i, is the Loga-
rithm of I etc I ; therefore that Number of thofe
RatiuncuU which lie between, or exprefs the Di-
flftnceof 2, 3, ^c. to i, fhall alfo be as much the
Logarithms
Chap.tl. OfmaiingaTaMeePLoQAiLiTHMt.^
Li^arisbm of thofe NumbQrs 2» j, 6fr. or, ntfacr,
of their Ratio's to Umty. Atid thus it appears that
even the Logarithms of 2, 3, 4, .5, fc?c. ^and for
the fame Reafon) 11, I2, i j, i^c. 21, 22, 23, £«?f,
loj, 102, 103, i^c. may be found to as gteat Ex-
adtnefs as is ncceflary. The Method of doing this
by Numbers, ihall be the Subjeift of the next Chap-
ten
C H A P. U.
Of the Met hod of making a Table of Logarithms
by plain Arithmetic.
I. T TAving explain'd the Nature and Properties
X i of Logarithms, and fhewn not only how
they are adapted to Series of Geometrical Pro-
portionals, but alfo by what means they are to be
calculated for all Numbers from Unity to any large
Number either above or below it :
2. Ths, Method or Procefs it felf, then, of doing
this, is the next thing to be propofed ; an ar-
duous 7ajk this, to him that firft aggre{s*d it ! At
fuchan Enterprize, he indeed might truly have laid.
Hie Labor, hoc Opus eji. And this any but an i»-
human Reader will be convinced of, by viewing the '
following tedious and laborious Procefs for gaming
the Logarithm of one fmgle Prime Number, and
that only to feven Places of Figures.
3. For, as I faid before, 'tis fufikient to produce '
the Logarithms of Prime Numbers only , becaufe, by
the Additioft, Subftradtion, 6fr. of thefe, we obtain
the Logarithms of all Compofit Numbers, with
Eafe ; as will appear further on. Now, tho* there
are but three Prime Numbers, viz. 3, ^, 7, between
C I
'io Of the Method oj making
I and 10 ; yet» becailfe the Logarithm of no Num*
ber between i and lO can be round by the Loga-
rithms of any of the Proportionals I, 10, IOO9 IOOO9
(Sc. it follows^ that the Logarithhi of any one of
the nine Digits is found with equal Difficulty for the
firft. Therefore I Ihall give an Example in finding
the Logarithm for the Number p.
4. The Number 10, then, bemg already fuppofed
at fuch a Diftance from Unity as contains 1 0000000
equal Parts or RafiuncuUi which is the Logarithm
or 10 ; therefore to find the Logarithm of -9, is to
find bow many ^ or what Numier oi xhok RatiuncuU
are contained between t and 9, which is done in the
following Manner.
5. Firft, make A=i, whofe Log. is 0.0000000 ;
and B=:io, which Logarithm i. 0000000 ; as in the
Table below.
Secondly, between A and B, or i and 10, find a
Mean Geometrical Proportional 0=3.16227775 the
Logarithm whereof will be half the Logarithm of 10,
l;/z. 5000000.
Thirdly, becaufe C is much lefsthan 9, find an-
other Mean Proportional D=:5.62 34 13 between B
and C ; whofe Logarithm will be an yfritbmetical
Mean between the Logarithms of B and C, vitt,
0.7500000.
Fourthly, becaufe D is ftill much lefs than 9, find
another Geometrical Mean between B and D, viz.
£=7.4989421,* whofe Logarithm is an Arithmetical
Mean between thofe of B and D, viz, 0.87500000.
Fifthly, fince E ii yet much Icfs than 9, find yet
another Geometrical Mean F=8. 659643 2, between
B and E ; the Logarithm of which will be an Arith-
metical Mean 0.93750000, between thofe of B
andE.
6. And thus continue finding Mean Geometrical
Proportionals between the Numbers next Greater and
ne^ctLeJfer than 9, till you arrive to the Number 9
itfelf.
Chap.II. a Taile ^Logarithms* tt
it/elf, which (hall be clear of all other Figures, but «
Cyphers^ to the Number of Places propof^ ; which
will happen at the twenty-fixth Trial, as you fee in
the following Table.
Proportionals. Logarithms.
A i.ooooooo
C 3.1622777
B 10.0000000
B 10.0000000
D 5.6234132
C 3.1622777
B 10.0009000
E 7.4989421
D 5.6234132
B 10.0000000
F 8.659B432
E 7.4989421
B 10.0000000
G 9-3057204
F 8.6596432
G 9.3057204
H 8.9768713
F 8.6596432
G 9-305>204
I 9.1398170
H 8.9768713
I 9.1398170
K 9^Q579777
H 8.9768713
K 9^0S79y77
L 90173333
H 8.9768713
L 90i73'333
M 8.9970796
H 8.9768713
C 2
o.oooooooo
0.50000000
I.OOOOOOOQ
1. 00000000
0.75000000 •
0.5000000Q
I.OOOOOOOQ
0.87500000
0.75000000
I.OOOOOOOQ
0.9375000Q
0.8750000Q
I.OOOOOOOQ
0.96875000
093750000
0.968 7500Q
0.953 1 250Q
0.9375QOOQ
0.96875000
0.96093750
0^953 '2500
0.96093750
0.95703125
0.953^-500
0.95703^25
0.955078 u
0.953 1 250Q
0.95507812
0.954IOI56
0.95312500
Ya Ofthi Method of making
Proportionals. Logarithms.
L 9017333?
N 9,0072008
M 8.9 9 707 96
N 9^0072008
O 9.0021388
M 8.9970796
G 9.0021388
P 8.9996088
M 8.9970796
?) 9^^0021388
Q^ 9.0008737
P 8.9996088
Q^ 9.0008737
K 9.0002412
P 8.9996088
K 9.00024x2
S 8.9999250
p 8.9996088
R
T
S
T'
V
S
9.0002412
9.000083 1
_8. 99992 50
9.0000831
9.0000041
8.9999250
y 9.0000041
X 8.9999650
S 8.9999250
V 9.0000041
V 8-.0000845
X 8.9 999650
V 9.0000041
Z ' 8.9999943
y 8.0000845
0.95507812
095458984
0.95410156
0.95458984
0.95434570
0.95410156
0.95434570
0.95422363
0.954 1 01 56
o 95434570
0.95428467
0^0542236^
0.9542 »4b 7
0:95425415
0.^5422363
0.95425415
0.95423889
0.95422363
0.95425415
0.95424652
0.95423889
0.95424652
0.95424271
9.95423889
0.95424271
0.95424080
0.95423889
■ !■ ■ III! II ■!#
0.95424271
0.95424217
0.95424080
09542427!
0.95424223
0.95424217
Chap.IL ^Ti?^/??/* Logarithms.
Proportionals. Logarithms.
»3
V^ 9.0000041
a 8.9999992
Z 8.9^9994?
V 9.0000041
b 9.0000016
a 8.9999992
b 9.0000016
C 9.0000004
a 8.9999992
c 9.0000004
d 8.9999998
a 8.9999992
c 9.0000004
c 9.0000000
d 8.q99Q9Q8
0.95424271^
0.95424247
0.9^42422^
0.95424271
0.95424259
0.95424247
0.95424259
0.95424253
0.95424247
0.95424253
0.95424250
0.95424247
0.95424253
0.95424251
0.95424250
7. Thus, after a long and laborious Calculation,
(confining of 26 Multiplications and Extradions of
large Numbers) we at length have obtained the Lo-
garithm of 9, viz. 0.95424251, which is cxacftly
true to 7 Places or Figures : That is, if the Loga-
ritbm^ or Diftance of 10 from Unity be fuppofed to
confiff of 1 0000000 equal Parts, then the Diftance
of 9 from Unity fliall confift of 9542425 of thofe c-
qual Parts precifely % which therefore is the Loga-
rithm of 9, becaufe it exprefles the Ratio of 9 to i .
Jf the faid Logarithm, or Diftance of 10 from Unity
were fuppofed to confift of looooo.ooooo.ooooo.-
00000. 00000. oocoo. 00000. 00000 00000 .ooooo.-
00000.000000 equal Parts ; then the Diftance of 9
from i,wouldbe95424.25094.30324.87459 00558^- .
06510. 23061.84002. 57728.38139.17296.59731^
of the fame equal Parts j and is its Logarithm to 62
Places of Figures. But the Reader is not to imagine
that this prodigious Logarithm was produced by the
I fore-
14 Of the Met hd of making
foregoins operofe Method : No ; for tho* 'tis not
impoffibie, *tis impraAicable by that Method -, but
it is the Produd of modern Invention -, an Inftance
of which will be given in due Place.
8. Having thus found the Log. of 9=0.9542425
if it be divided by 2, the Quotient or Half, 0.477 12 1 2
will be the Logarithm of 3, by the Converfe of Ar-
ticle 12. Chap. I. The Double of the Logarithm
of 9 is the Logarithm of 8 1=1.9084850 by the fame
Article. And thus by a continual Multiplication of
this one Logarithm by 2, 3, 4, 5, (^c. you gain the
Logarithms of all the Powers of Nine.
9. 'Tis farther evident, that if the Ratio of 10 to
I, 100 to 10, 1000 to- 100, 6?^. be fuppofed only
Unity or I, as exprefled in the Table of Art. ij.
Chap. I. the Logarithm of Nine, viz. 0.95424225
will be jio other than a Decimal Fraftion, whofe De-
nominator is the faid Ratio^ or Unity, with Cyphers
annexed, thus tlUllll .
^.8 *" 3=047712*2] g
9=0.9542425
10=1.0000000
81=1.9084850 >
100=2.0000000
729=2.8627275
1000=3.0000000^
j[-iP-««/. 10000000
q=0.-Ali24ii
J7"~"^* I 0000000
10=1.0000000
A — * •xooooooo
100=2.0000000
/^y— ^•looooooo
^ 1 000=3. 0000000
10. Hence 'tis obvious that the Logarithms are
only the Decimal Parts of feveral Ratio^s^ i, 2, 3,
&?f. of 10, 100, 1000, tfc. to Unity ; and that
thofe Ratio's themfelves make the integral Part of
the mixt , Decimal Logarithm, (as I may call it.)
Thefe integral Parts of thofe mixt Decimal Loga-
ritbmSy then are what we call the Indexes^ or rather
Indices^ of thofe Logarithms, or fractional Parts \
wherefore, the Rations of lo, lOO, 1000, 1 0000,
1 00000, &?r. to Unity ^ are the Indices of or belong--
ing to the Logarithms of all intermediate Numbers im-
mediately
Chap. II. tfTtf^Zf^ Logarithms. 15
mediately above ihcfe Proportionals. So die Ratio of
10 to I, viz. I, is the Index of the Logarithms of
all Numbers between 10 and 100. And 2 (the
Ratio of 1 00 to i) is the Index of all Lc^rithms of
Numbers, from 100 to looo^ and (b on.
11. From whence 'tis eafy to obferve, Thai the
Index of any Logarithm contains a Number of Units
lefs by one than is the Number of Figures in thaiNum*
her which belongs to the f aid Logarithm.
Therefore f g^.
12. Now as this Index is thus called, from its/x-
Heating or fhewing how many Places of Figures
there are in the Number of the Logarithm ; fo, oa
the contrary, the Number of the Logarithm, as
plainly ihews what the Index of the Logarithm muft
be. And therefore, fince in the Tables ofLogarithms,
all the Numbers of thofe Logarithms are appofitely
placed in proper Columns by them, *tis entirely need-
Icfs to print the Indices^ as being well enough known
by the Numbers ; and thus they are (for this Rca-
fon) omitted in feveral of the faid Tables ; and con*
fequently in thofe we have here, info concife a Fotm^
now firft of all made publick.
13. What has been faid hitherto, relates altoge-
ther to the Logarithms of Whole Numbers ; but the
fame DoStrine is equally applicable both in Theory and
PraSice^ to fraaionai Numbers \ for in EfFeft, the
Properties of Decimal and Jin;>ole Numbers are the
feme. Thus the Series of decuple Proportionals ^ i,
10, 100, 1000, 1 0000, 6fr. may be continued as
well belowVviitj^ as above it -, for the following Rank
of Numbers, viz. rdurs, ttoE^j irs* /ny> i» i^t 100,
I GOO, 10000, i^c. which may be thus exprefs'd,
0,0001, 0,001, 0,01, 0,1, I, 16, 100, 1000,
lOOOO,
1 6 Of the Method ofmakwg
loooo, lie. are all in a Geometrical Decuple Ratio
tr Preportion 5 and confequently equally diftant from
Unity on each Side, and io their Logarithms eqiii-
differcnt, and the fame. The Ratio's then, in the
Scale above Unity, may be called Pofitive Ratio^s^
and thofe below Unity, Negative Rations i and thus
the Index of a Logarithm of a ^i&^/^ Number^ or /«-
ieger^ fcall ht fcfiti'oe^ but the Index of the Loga-
rithm of FraSian ftiall be negative^
14. Andiince Vulgar FraStions^ Duodecimal^ Sex-
cgefimaU &c. Fraftions, are all reducible to Decimals ;
*tis evident that all Numerical Arithmetic whatfoever
is fnbjeft to, and manageable by this Art of Logarithm
mical Arithmetic. The only Difficulty being in ad-
apting or readily finding the proper Index to the
Logarithm of a Decimal Number. For, fince the
Index in the Logarithm of Integers^ fhews only the
Number of Places or Figures m the faid Integers, *tis
plain thofe Indices decreafe^ as the Places of Figures
in the Integer decreafe, and intirely vanifli when
thofe Places of Figures become Unity or one, that
Js, the Index is then o ; and confequently canndt
ferve for Decimal Numbers.
15. Therefore fome new kind of Indices or Cha-
^ raSlerifiics muft be invented, which ihall be proper
only to Decimal Numbers^ as the other are to Inte*
gers ; and fuch as (hall as readily difcover the Num^
ber of Cyphers to be prefixed to thcjignificant Figures
m the Decimal^ as the other determine the Number
of Figures in the Integer^ are in the fame degree ufe-
ful for Decimals^ as they for Integers. For as to the
figi>ificant Figures of the Decimal^ the Logarithm it
felfdiCcovtrs them ; all therefore that is farther- ncceA
iary, is to procure fuch Indices as fhall at all times
denote how far ^ or how many Places froei Unity ^ the
firfl Jignificant Figure of the Decimal muft ftand ; or,
which amounts to the fame, how many Cyphers mut
be prefixed to compleat the true Value of the Decimal.
16,
Chap.lL a TaiU '^Lo GAR ttuMil t^
c^y 500=4.6^^66936
Integers^ 4750=3.6766935
t 4';S=2.6y66g^6
47,5= 1.6766936
4,75= 0.676693^
yb475:=z.i. 6766936
y0047$=:.7. 6766936
^0004.7 5=2.6.6766936
,00004.75=. $.6766936
It
&
16. Tht Indices {qx
this JPulrpofe am there-
fore judg'd beft, Wi&iVi?
i^/j8f^ fubfiraSted from
9, ^/i// i&aw a Re
mainder expreffing the
Number of Cyphers to
be prefix" d'^ thus the
Index o( a Decimal
who(e firft Figure (to
the Left,)isfignificant, i
ttiuft be 9, becaufe 9 — 9=0, tha^ is nd Cypher is ta
be prefixed* ^ And thus, a Decimal thaj; has i^ 1, 3^
4, &ff. Cyphers^ muft have the Indices^ accordingly
8, 7j 6j 5, fcf^. becaufe 9 — 8=1, 9 — 7=2*9 — 6=3,
9 — 5=4, (^c. denoting the Cyphers to be prefixed to
the Decimal Numbers. This is tftridcnt by the Ex-
amples in the Table above ^ but it is proper to dott
thefe new Indices on each Side^ to diftinguifh 'em
from thofe which belong to the Logarithms of//r-
tegers.
17. From all that has been faid hitherto, *tis evi-
dent, that while th^Jignificant Figures in af^ Number
ivhatever^ Integral or Decimal, are the fame, the La-'
garithm of thofe Figures will be the fame alfo ; the
Variation occaiion'd by the differing Nature of the
Number, being only in the "Indices^ as denoted ia
the faid TaBle. But fomething farther will be faid
of the Nature znA Variety ofhdices^ when we com^
tojireat of the PraSiical Rules oi A^diti/>n and Sub^
firaSion of Logarithms ; I fliall now proceed to the
next Chapter, wherein the ^erious iVatun^ or Doc-
trine of Logarithms will be farther illuftrated, and
rendered more obvious to the Scnfcs by Geometrical
Schemes and Demofffirations.
D
CHAP.
i8 Vfle Nature and Properties
CHAP. III.
iTbe DoSirine of the Nature and Properties ofLo^
garitbms farther explained and illujirated^ bf
means of tbe Log ARiTUMic CvRvt.
i.T N explaining the Nature of Logarithms from
X the Logarithmic Curve^ *twill be expedient to
Ttprtknt Proportional ^antities by Letters or Species ;
fis being fuited to a more univerfal Theory than Ntm-
bers^ and better applicable to Right lAnes^ by which
alone both Numoers and their Logarithms are repre-
fcnted in the Geometrical Method now before us.
2. In a Series of ProportipnalSj then, increafing
from Unity, let the firft be ==tf ; then will the Se-
ries be I, ^% a* J a\ a\ tf% &c. where /tis evident
the Indices or Exponents of the feveral Powers of tf,
are a Series of Numbers in Arithmetical Progreffion^
each, whereof fliews the Place or Diftanee of its TVnw
from Unity i thus the Term a^ is (hewn by its Index
4 to be in the fourth Place from Unity ; and a^ is in
the Jifth Place ; or a^ and tf' are four ^nAfive times
snore diftant from Unity than the jf^^2Vr»» a ; which
is her^ the common Ratio alfo.
3. From hence you obferve, that the Exponents
of the Powers of the Terms in the Series i, a*, a\
a\a\ a*^a\ a^j &c. are the Logarithms of thofe
Terms rcfpedively. Let the Exponent be =^ ;. then
c the Terms, i, «, aa^ aaa^ aaaa^ aaaaa^ aaaaaa^ Sec.
I theExpon. o, ^, 2^, 3^^ 4^?, 5^, 6e^ &c.
Therefore
c As aaxaaa=a* 5 aya'=a^ ; and aamacf^zra'^^'a^
|Sq 2^4-3^=5^; H-5^=:6^ ;.and 2e+3ie=zgie.
and fo every where, agreeable to the Nature of 2>.
garithms
Chap.III. g/LoGAniiTHMS explained. 19
garitbms beibre defcrib'd, in the two precedingChapv
ters.
4. If between i and a there be put one mean Pro-
portional^ viz. ^?j its Index ot Exponent muftbe
wrote i ; thus ^a:=:a^ \ for o, i, i, have an Ariib^
metical Ratio ; and fo the Terms i, ^, a^tat Geo*
metrical Proportionals. Thus» a mfm Proportional
between a and tf^, is ai^assa^i=o^. Al(b if you
conceive two mean Proportionals between i and a^
they fhall be a^^ a^ ; for i : a^ :iinia\ ando, i, $9
I, have an equal Difference.
5. Moreover the feme Series of Geometrical Pro*
portionals may be continued both ways, and be made
decreafing as well as increajing ; that is, it may as
well defcend below Unify to the left Hand^ as afcend
above it to the Right. Thus the Terms ^t, ^i, ^t,
^iy 49 I9 ^9 ^% 41% tf^ ii% &c. areall in theiame
Geometrical Progrejfion. And fince the Diftance of
a* from Unity towards the right Hand is pofitive^nnd
denoted by the pojltive Exponent -Ht ; fo n^ being
equally diftant on the contrary Side^ or helot» Unity^
and this Diftance being negative^ therefcH^ the Ex-
ponent thereof may be thus negatively expreis*d, '
a^^ z=a:^ I, and fo d^ is the fame as ^~^ ; thus the
Series tf~^ jJ~^ iT^, a~^^ a~\ U «» 4% a\
a\ ^S &c. is the fame as the laft foregoing.
6. Therefore, in thoie 5ieries, if a repreient any
Right Line (fee Fig. i.) aaj or the Square thereof,
is not to be taken as a C^antity of two Dimefj/lons^ or
Surface^ viz. a Geometrical Square ; but only as a
Line that is a third Proportional to fome Line taken
as Unity ^ and the Line a. So likewife aaw^ or aaa^
is not a ^antity of /i&r^^ Dimenjions^ or a Geometric
cal Cube 5 but a Line that the fourth ^erm or Pro*
portional in a Geometrical Series^ whoie j&;t ?Wtii is i *
D 2 and
5.0 The Nature and Properties
and fecond a. And thus you are to conceive of al!
the reft; viz. limply as Proportionals of Length
pnly.
7* Thi^ bein^ premifed t|ien, if on any Line, as
AN (Fig. 2.) both ways indefinitely extended from A,
be taken the equal Diftances Ae=BCE=EG3=GI=3
ILsLN, to the right Hand \ and Acs=ce=eg, ^c^
on the left ; and if on the Points g, c, c, A, C, E^
G, I, L, N, be ereifted to the Right Line gN, the
Perpendiculars gh» ^, cd, AB, CD, EF, GH, IK^
LM, NO, which I^t be vtx a continual G^ciw^/r/V^/
Proportion to each other, and reprefent Numbers^
whereof AB=i, or Unity. The Tops of thefe
Lines being duely join*d, from what is call'd the
Logaritbptetic Curve^ viz. hBHO 5 by which we
are farther to explain and illuftrate the Nature and
Properties oi Logarithms.
8. But a more com pleat Idea of the Logaritbmical
Curvey may be conceiv'd by a twofold Mption of the
Line AB ; the one equable^ the other accelerated or
retarded in any given Geometrical Ratio : For Ex-
ample, if the Right Line AB moves uniformly along
the AN, fo that the End A defcribes'^^w/j/ Spaces
AC=:CE=?:EG, Gf r. in equal Times ; while in the
• fame Time the faid Line AB fo increajes^ that the
Increments thereof generated in equal 'Times ^ be fro^
pcrtional to the whole increafing Line ; that is, if AB
in moving forward to <z^, be increafed by the Incre^
ynent ob^ and in an equal Time^ when it comes to CD^
the Increment thereof is jyp ; and Dp : ^^ :: ^^ : AB ;
then the End B of the faid Line AB thus continually^
increajing or decreajing in xhtfame Ratio^ will defcribe
the Logarithmic Curve. For fince AB : ko :: ab : Dp
:: DCidq^ &c. it Ihall-be, by Qompofition of Ratio,
a3 AB lab y.abi DC :: DC : dc ; and fo on.
9. Since (from wjiat has been faid) the Line
AB=i , and the other Lines CD, EF, GH, iSc. pro^*
cced ftom theiKC to increase in a Geometrical Ratio^
and
Chap. III. ofh OGARiTHMS explained. %i
and their Diftances AC, CE, EG, Csfr. are all efual
to each other (by Art. 7.) Therefore it follows that
the Diftances or Lines AC, AE, AG, ^c. are the
Logarithms of thofe Numbers reprefented by the
Right Lines CD, EF, GH, 6?r. according to tho
Definition of Logarithms ^ Chap. I. Art. i, and 13.
For if AC=i, then AE=2, 'AG=3, (<fc. and fo in
the Serits of proportionate Numbers AB, CD, EF,
GHj IK, LM, NO, l^c, we have the Logarithms
o, AC, AE, AG, AI, AL, AN,j.,
o,iAC, aAC,3AC,4AC, 5AC, 6 AC, J^^*
10. As Logarithms^ then, are the Exponents of the
Rafters of proportional Numbers to Unity in any Se-
ries 5 or ihew the Place^ Power ^ or Order of the.
Numbers with refpeft thereto; 'tis plain that the
Logarithm of AB or Unity muft. be =^, becaufe U-
nity is not diftant from it felf. Alfo if the Ratio of
CD to AB be z=:x^ then fhall the Ratio of the Num-
ber EF to AB=x X K=xx ; or Duplicate of the former 5
and the Ratio of GH to AB=xxxx;if=;f', or Tripli-
cate of the firft Ratio x. Thus the Ratio of NO to
AB=i;f^/ And confequently the Numbers^ their Ra^
tio^s^ and Logarithms^ will ftand in the following
Order, viz.
Prop. N°. AB, CD, EF, GH, IK, LM, NO, 6fr.
Ratio's ... I, AT, ;v*, ^% x% ;?% x^^ tfr.
logarithms o, AC, 2 AC, 3 AC,4 AC,5 AC, 6AC,&?f .
11. If four Numbers be fuch, that the Diftance
between t\it firft and fecondy be equal to the Diftance
between the third and fourth, (be the Diftance bt-
tyvtcn the fecond and third what it will) then thefc
Numbers will be proportional ; for let the four
Numbers be AB, CD, LM, and NO ; then becaufe
AC=LN, it will be AB : Dp :: LM : OT, by Art. 8.
Therefore (by Compofition of Ratio) AB : DC :: LM
: NO. And thus of any other four Numbers in the
Geometric Serie^^. The Converfi alfo of this Propo-
iition i? a3 inanifeftly true.
12.
22 ^['be Nature and Properties
12, Since ABrai, and its Logarithm =o ; and
fince Unity is to the Multiplier as the Multiplicand is
to the Predulf, in every MultipJication ; therefore to
every Addition of Logarithms j there correfponds a
Multiplication of their Numbers. Thus,
Number, CDxEF=:GH ; and EFxIK=NO,
Logarithms,AC+AE=AG5andAE+AI=AN.
And in Divifion^ fince Unity is to thtDivifor as the
Siuotient to the Dividend % then (by -/fr/. ii,) for
every SubfiraRion or Difference of Logarithms^ there
correfponds zDivifion of their Numbers, Thus,
Numbers, GH-r-EF=CD; and NO-^EF=IK.
Logarithms, A(ir-AE=AC ; fo AN— AE=AI.
agreeably to what was fhewn in Chap. I. Art. lo.
i^. Again, fince Unity y any affum^d Number ^ the
Square thereof, the C«^^, the Biquadrate^ &c. are all
continual Proportionals^ their Diftances from each
other will be ^jf«^/ to one another ; and are therefore
as the Numbers AB, CD, EF, GH, 6f(r. in the Se-
r;es. Confequently if the Diftance or Logarithm of
the Number CD be multiplied by 2, 3, 4, fcfr. there
will anfwer an Involution of the faid Number to the
Square^ Cube^ Biquadrate^ &c. Power.
Thus for the Square
Numb. CDxCD=:EF ; and EFxEF=IK.
Log. ACx2=AE5fo AEx2=Al.
For the Cube
Numb. CDxCDxCD=:GH -, and EFxEFxEF=Nb.
Log. ACc 3 = AG ; and fo AE x 3 =AN.
For the Biquadrate
Numb. CDxCDxCDxCDC=EFxEF)=IK.
Log. ACx4=: ( AEx2=:) AI.
Which is the lame as was obferved in Chap. I.
Art. 12.
14. If the equal Diftances AC, CE, EG, GI, IL,
be bifcficd, and in the Points of Bifedion /ar, r, e^g^i^ -
there
Chap.in. of Log Animus explained, aj
tiiere be erefted the Perpendiculars ai^ cd^ //, gb^ ik i
thefe, by means of the Curve, will be all Proparti^^
nals ; and the Number LM will be in the tenib
Place from Unity or AB : if then we put LM=io»
and fuppofe its Ratia to Uniiy be looooooo, fudi
RatiuncuUf as that of n ^ to AB, or Unity b i.
iThen will the Numbers and their Logariibms be «•
exprefi'd in the following Table.
Numbers.
AB=i
Logarithms.
0.
ab =1.259, t^c.
Aa = loooooo
€0 = 1.585, £s?r.
AC= 2000000
'
cd = 1.996, fcfr.
EF =2.512, 6f^.
ef :=: 3.163, (^c.
GH= 3.982, £sfr.
Ac = ^000000
AE= 4000000
Ae = 5000000
AG= 6000000
|i& =5.072, fcfr.
IK =6.310, &ff.
A^ = 7000000
AI = 8000000
i* = 7.944, 6fr.
LM=io.
At = 9009906
AL =10000000.
i
fJumiiers.
Logarithms.
AB = I
CD = 2
EF =3
AC= 3010300
,AE= 4771212
GH = 4
AG= 6020600
IK =5
LM= 6
NO= 7
PO = 8
RS =9
TV =10
AI = 6989700
AL= 778151^
.AN= 8450980
AP = 9030900
AR= 9542425
AT =ioooooQO
15. If BKV (Fig. III.) be the Logaritbmetie
Curve, and AB=i, TV=io, or 10A6; and all
Ae intermediate Lines CD, EF, GH, fs?r. be as
Digits 2, 3, 4, fcf^, to IP i thefe Lines will befitu-
ated
2 4 iie Nature of LoGAHtTUMs.
ated at unequal Diftances from each other ; and fo
€titit Logarithms ACy AE, AG, fe?r. not proceeding
to increafe by equal Differences, (hew the Numbers
AB, CD, EF, 6f r. are not ordinately (or all of them)
io a Geometrical Ratio or Proportion. If the Ratio
of I o to I, or TV to AB be fuppofed to confift of
looooooo RatiuncuUj and confequently the whole
Logarithm AT of iooqoooo equal Parts ^ then the
Diftances of each of the nine Digits from Unity, viz^
AC, AE, AG, £sfr. fhall confift of fuch Numbers
of thofe fmall equal PartSy as are expreffed in the
Table, oppofite to the faid Digits.
-1 6. Yet fince fome of the Digits are in a Gebme^
trie Ratioy as 1,2, 4, 8, their Logarithms will be
equidifferent ; fo AC==CG=GP ; and becaufe 1:3::
3 : 9, there is AEssER ; and thus it appears that
tho' the nine Digits are not all in continual G^(?iw^/r/-
cal Proportion^ yet fome of them are fo ; and the
reft are fome of thofe Proportionals^ of which there
be looooooo between AB=i, and TV.=io. If
the frfi Term from Unity be called Xy the fecond
will be x*y the third ^% &c. and fince. the Number
10 is the 1 0000000th Term of the Series, it will be
TV=^'°°^°°°^=io, Alfo CD=x^°'"'"°=2; and
EF=^"''"*"=3 ; and fo on. Whence every Digit
is fome Power of that Number which is th& firft Pro-
portional from Unity : The Exponents or Indices of
the Powers being the Logarithms of the Numbers^
agreeable to Art.'io. of this.
^
CHAP.
C H A P. IV.
Tie Nature of Logarithms and their Indiceil
ivben the Numbers are FraStionSy farther ex^
plairid by the Logarithmic Curve.
i.^TTE have hitherto principally confidci*cd thft
VV Nature and Properties of Logarithms of
vohole Numbers or Integers^ and have obierv^d that
in the Decuple Series t, io, loo, iooo,,€^f. the
Terms have their Rations to Unity iMrmative or pth
fitivCy viz. I, 2, 2, 6ff. orthu$, \i^ 4-a, H-j^
&r. the contrary of which happens when tne Num-
ber of the Logarithm is not integraU but fraHional^
or cxprefles only fome/ri?^i^»tf/^ Part of Unity. FoJf
there the Series beiilg continued on the other Side^ or
below Unity ^ hzxh tht Indices of the Powers of the
Terms of a polity direftly oppqfite to the former |
. And therefore as tbofe Indices were pcfittve^ to tbefe
will be of a negative Nature^ and import the Terms
to be below the State of Unity ^ or rather of integrity i
and willbeafieded with the' Sim --»» as>«»i» — ^2^
— 3, 6?r. as is evident Chap. IIL Art. 5.
2.. Wherefore fince AB reprefents Unity ^ (Fig. 11.)
all the Numbers in the Series towards the right Hand^
or above Unity ^ CD, EF, GH, fcfr. being integral^
and having the Ratio greater than Unity ^ will have
the Logarithms thereof pqfitlve^ viz. -f-AC^ -l-AEt
+AGp &c. But thofc Numbers or Terms on the
Lefi^ or bck)W Unity, cd, ef, gh, 6?f. being /w^-
tionate^ and having the Ratio lefs than Unity or if-
creafingy will have the Logarithms thereof Negative,
V/z. —Ac, -—Ac, -— Ag, €g?r. Andfothei;»;i;V^j
E qI
26 Tie Natuf^ebf Log Ai^iTfiMS^ an4
ofthofe Logarithms will be refpeAively affirmative
or negative.
3. Since then (as hath been all along (hewn) the In^
dices of the Logarithms of Integers^ as bemg affirma^
tive^ muft be addei^ that fo their ^um may indicate
the PrcduEt of the Numbers multiplied ; as AC+AE
s=AG, which fticws that its iV«iwii?rGHfi=sCDxEP,
the Numbers multiplied ; fo if the Index of one Lo-
garithm be negative^ and tht Index of another htaf-
firmative^ the Difference of thofe two Indices muft
be taken for th& Product of the Numbers. Thus to.
multiply the Integer GH by the Decimal cd, their 7»-
^/V^j being H-AG> and — Ac, their Difference
AG — AssAE, and fliew the Number EF is the
ProdflEt of the other two. And here, becaufe the
greater Index GH is affirmative^ the Difference alfo
AE is affirmative^ and the Product EF to be an In-
teger or on the right Hand of Unity.
4. But if the Decimal gh be to be multipliedhy the
Integer CD, whofe affirmative Logarithm or Index
AC is the negative Index of the Decimal^ viz.
Ag» their Difference Ag — AC = Ae, is negative
alfo, and fo ihews the ProduSltf will be a Decimal^
or ^e/^w Unity. Again, if both the Indices of Loga-
rithms^ whofe Numbers are to be multiplied^ are ne-
gativej their 5tfw ftiall be a negative Index whofe Lo-
garithm points out the ProduSi (in this Cafe) always a
Decimal^ or in the Series Wewc; Unity AB, For Ex-
ample, to multiply the two Decimals cd, ef, the Sum
of their negative Indices Ac+Ae=Ag is negative^
and fhews the Decimal Pr^dullgh^ ever below Unity.
5. The Reafon of all which is very plain; for
fince Unity is to the Multiplier as the Multiplicand to
the ProduB^ and the Logarithm of Unity is =0 ;
Unity ^ any /te;^ Numbers and their ProduSt^ together
with their Indices^ will ftand as follows :
Chap.IV. their Indices^ farther txplami, 27
, cind. 0, AC; AE, AG,
'• iNum.AB:CD :: EF : GH,
'1
M
iCD X £F=GHx i=GH.
o, —Ac, +AG, -l-AE,
. AB : cd :: GH ; EF.
o, '+-AC, — ^Ag, — Ae,
AB : CD :: gh : ef»
♦i«H^fo«. C AC— Ag = Q — Ac = — Ae,
thcrcfort JcD x gh = i x cf = cf.
CO, —Ac, —Ac, — Ag,
♦•t AB : cd :: cf : gh,
therefore }^?^^^ = ^~\« = -A&
C ef X cd =s I X gh = gh.
6. FrQm whence 'tis evident, if the Numbers are
hotb Integers^ the Produ£I will fall in the Scale above
for be greater than) eithpr ; if they be of different
Sorts^ the Produff will fall between them, above
Unity if tht greater Number be an Integer^ orbehw.
Unity ^ if it be a Decimal. Laftly* if both the Num*
bers be Decimals^ the Product will fall below Unitj
and them botb^ or will be lefs than either of them.
Thefe things well obfitrv^d and underftood^ make all
the Myftery of the 4ritbmetic of Fraai(ms by I^^-
r;/ibij vaniih, where it is taught with the Ufe ofne^
gative Indices to Logarithms oifraStional Numbers^
as it is to be found in fome Books. Note^ \ have all
along in this Chapter, faid affirmative or negative In*
dicesy and riot affirmative and negative L^aritbms i
becaufe the Indices only (hew the ^ality of the
Numbers^ viz. whether above or belowUnity^ greater
or leffer than it, i. e. whether they be Integers^ or
wholly Fra£f ions i and therefore theCe Indices iqvA be
carefully added or fybjiraj^edy a^ o^cafion requires.
E 2 and
90 ^ ffature ff Lo g Ami t h m8, and
and as before direAed ; but the Logarithms fhewing .
th^ Difiances o( Numhrs from Unity indifferently^
or without rcfped to thtOrder oi akov^ or helow^ are
always to be. added.
7. But fince this Method of negative or different
Indices to Logarithms is attended with Mditiphj Sub^ ^
JiraStion^ and other intricate CautionJi peculiar to
themfclves^ it can't be recommended Co much asan^
other Method more approv*d^ and therefore more ge^
nerally ufed\ whefein th^ Indices of all I^ogfritbms^
(both offraSional ^ well as inSegral Numbers) un-
dergo the f^meco^xnon ^oMagement with their X0-
garitbms,\ and fo can*i he fo difficult to Learners as
the £?/i&^r ; tho' it is alfo attended with particular
Rules ^ as you'll iiqd farther on, and i$ as foUows :
Nimibers.
Indices.
Lor- 1
ut =,0000000001
— 10 90.0000000
rs = >ooQOOOooi
tr = — 9 I. 91.0000000
''
pq=: ,00000001
tp = — 8. 2. 92.0000000
1
no =2 ,0000001
tn = — 7. 3. 93,0000000
lm=;;: ,000001
ife =; ,00001
tl=: — 6. 4. 94.0000000
ti=; — 5. 5. 95.0000000
^
gh:;;: . ,OOOl
tg=: — 4. 6. 96,0000000
efss ,901
\:e=-^ 3, 7. 97.0000000
cd=;, ,01
tc== — 2. 8. 98.0000000
abs 9I
ta=::-T- I. 9. 99.QOOOOOO
1
ABs? I
t A 5= 1 0. 1 00.0000000
I
CD:;; 10
tC = + 1 . 1 1. lOI.OOOOOOO
EF= 100
tD=;4- 2.1 2. 102.0000000
m
8. Let AB be ==?, or Unity (F%. IV,) and let
tu be ifraSlionai Number ip times as far below U-
fiity A3B^ as CD is ab&ve it ; then will AtzziqAC,
ima fiippofing the Number CDs=.io, ^is evident
te;5iooor6QQQ0Q> ^ f0999OpOO0i J if then, inftea<l
Chap, i V. their It^ctSy farther explained. 39
of the Logarithms beginning at Unity AB (as in
the former Cafe) ^i^e make 'cm beg^ from tu=
j^coooooQOOi, then the /»</^x of L^arithm of the
praaim rs^ will be i 5 of /y, 2 1 of i»» 3 ^ and ib
of Unity AB> 10 ; and iinceCDsio, £F=xoo ; the
Indices of their Logarithms will be tC=5ii, and
tD=i2, ^c. as is evident in the foregoing Table ;
wh^you ke tht fraaicnal and integral Numkers
with Indices oi three feveral Kinds fitted to the Lo-»
garithms of the faid Numbers.
• 9. That every thing might be made plain, I have
made the Index of the Lc^arkhm cS Unity ^ zsOt in
thtfirfi Rank of iHdiees ; and according (as we luive
iben' above) the Indices of all the Logarithms of
Numbers ab$ve AB will h^pqfitive^ as -hi, -Hi, (^c.
and of thofe below it negative^ as — i, —2, —3;
6f r. as in the Table. But if (as in the fecond Rank
of Indices) the Index of the L-ogarithm of Unity AB
be made =10 ; then (ball the Logarithms begin (as
before fuppofed) from the loth Place ielaw it, viz.
from the Number ut=,ooooooqoo I, sind then all
the Indices of all t\x^ /rational Numbers between
f bcoooooooo Of 0,0000000001 and I, will be ^r-
mative OTpJitivCy and not negative as before ; as is
fcen in the Table. So that by this Method we have
got pofitive Indices for the Logarithms of Decimal
Numbers as well as for Integral Numbers ; and fa
both ca{)able of the fame Management, which could
not be with negative Indices^ as before fhewn.
10. Alia in the third Rank (which is^ here gtveti,
not for Variety only, but hecaufe 'tis oftentimes of
Ufe) the Index of the Logarithm of Unity AB is made
=100 ; and confequently that thefe Logarithms be-
gin from the FraStion which is the 100th Proportion
nal below Unity, in the Ratio of r to 10, or -ti to u
Whence the Indices of Logarithms of all FraSiions
between o, oooBoooooooooooooooo 0000000000-
9OOOOQOOQOQOQQOOQQOQ OOOOOQOOOOOOOOOOOOOO
OOOOOOOQOO
30 The Nature ofLo GAR i t h m s, anJ
oooooooooooooooooooooooooooooi and i, will
of courfe hcpqfitive ; and fo in thisClafs, the Indices
oiLogaritbms of all decimal Numbers (greater by far
than^, or evenCuriqfityj candefire) will bepofitive
alfi), as well as thofe of the Logarithms of integral
Numiirs ; and therefore in (hort, we have obtained
fqfitive Indices for the Logarithms of all Numbers in
general^ and fo the Trouble of Addition and Subjirac^
Hon di different Farts of Logarithms at the fame
time, is avoided, as was propofed.
11. But (as I have before obferved, Chap. 11.
Art. 1 6.) the Indices of the Logarithms oi Decimals
ihould be diftingui(hed from the Indices of Loga-
rithms of Integers in Operations ; and this I think is
beft done as there dire&ed, viz. by fixing a Dott on
each Side the faid Indices. And thus manifeftly ap*
pears the Reafon of all that is delivered in Chap. 11.
from Art. 13. to the End.
12. Now in multiplying^ dividing^ &c. of Frmc-
tionsy the FaSors may be either both Decimal ; or one
DecimaU and the other an Integer \ if both the Fac^
tors zrt Decimal^ the ProduSl will bt Decimal^ in Mul-
tiplication. If the FaSors be of different Sorts y and
the Ratio of Unity to the Decimal be either lefs^
equal /^, or greater thap the Ratio of Unity to the /»-
teger ; then ihall the Produdb be greater^ equal to^ or
i^s than Unity \ i. e. it will be Integer^ Unity, or De^
cifhaly in multiplying ; all which will be evident by
the following Examples^ ferving as fo many Rules
for the right ordering of Indices m Operations.
13. TAB : cd :: ab ; ef ") «^- T cd x ab=i x ef=;:«f.
\ tA. .tc. .ta. ,te. > % v.tc.-f-.ta.=tA-+-te.
^'i 10. .8. .9. .7. J-S j.8.-K9,=io.'+-.7.
CThat is . , . . . 8,.H-.9. — io.==:.7.==:
Index of ef.
Chaf). IV. their Indices^ farther explained. 3 1
ab :: EF : CD! ^ f abxEFsi xCDsCD.
tE. tCA § ^ .ta.-HE=tA.H-tC.
12. ilJ-S L.9.+i2.=io.-t-ii.
nfcqucndy 9.+12. — 10=1 i.ae
Index of CD.
rAB:ab::CD:AB1u: fabxCD=ixi=i
3 tA. .ta. tC. tA. S g ^{ .ta,-HC=tA-f-tA.
3-^ JO. .9. II. 10. J -S 1^.9.-4-1 i=icH-io.
^Therefore »•..<♦..* .9.+11. — losssios^
Index of I.
Index of ab.
14. From thefc Obrefvations Well cOhfider*d, •tis
cafy to apprehend the Truth of what is delivered in
Art. 1 2 . above. And Unce AB, in thefc Examples of
the ProduRSy hath for the Index of it8 Logarithrti lO^
*tis equally obvious what the Indices of tht Logarithms
of the Products would be, wert the (aid /xr^^;^ of the
Logarithm of AB made 100. In iHtixsprefent Cafe
were tAsssio, if we rejcft 10, the Indices of the
Logarithms of integral ProduSls will be the fame as
if the Logarithms began at AB or Unity 5 as is plain
in the 2d and 3d Examples \ and accordingly if
tA=ioo, we rejeft 106, and the Cafe is the fame.
15. But fince 'tis moft convenient to have the In-
dices of all Logarithms of integral Numbers to begin
from Unity in the fimple Order, o, I, 2, 3, (^c. as
if the Logarithms did really begin from thence; fo
*tis but rgedting 10 from the /aid Indices In their
prefent State, dnd what we defire is obtained.
Thus
32 TbeNaiun df Log AJBitTUMs^and
ft.
C* 1 2.
•J* ^ .9. — lO ar ,7,
Thus the forc*3 2. .9. -h 12. — • 10 «= 1 1.
going Cafes 1 3. .9. -f- 1 1. — 10 = 10.
(^4. ii*-f-. .8, — 10 =zr .9.
by rejedHng 10 from r.8. -+- .9* — o = .17. .7.
the /»ifV^i of thej.g. •+- 2 — '0= ix. i*
Logarithms of/»-i .9. + i •^- o =5 10. o.
tegers^ becomes (, i 4- 8 — o :^ ,9. .9.
Then from the Sums •17, 11. 10. .9. again rqefting
10 (where 'ti^ found) we have the Remainders .7. i.
6. .9. the true Indices of the Logarithms of the Pr^-*
^ir^jy tsrequiredi
16. If either PaSloTy or the ProduSl of them^
exceeds the Limit of ,o'oo>oooooo » <^r is lefs than
^ooocoooooi, we (hall find it moft convenient to ufe
the Indices of thofe Logarithms of which the Indeoi
of the Logarithm oi Unity or AB is =-100, viz*
thofe -Indices in the 3d Clafs in Table of Art. 8. here-
of. And if, in all the foregoing Cafes, inftead of r^-
jeUing 10, we now rejeft 100, we (hall have the /«-
dices of the Logarithms of the Produds the fame
kind as before.
Thus^
byrejcfting 100,
becomes
.98. + *99. — 100 = .97.
.99. 4- 102 IQO = lOI.
'99' + ^^^ — ^00 =5 lod
loi.-f- .98. — 100= .^g.
.98. -f* ,99. ' — o =.197.
.99. + 2 — o = lOI.
•99* + I — o = loOi
loi. 4* •98. — o = 199.
•9^
I.
G.
*99-
Thus here you fee the EiFe<3: the fame as abov^t
Art. 15. if .97. .99. be deduced from 100, as .7.
.9. were from 10, the two middle Indices being the
fame in both Cafes ^
?7-
chap. IV. their IndkeSy farther explained. 33
J 7. In Bivifion^ the Divifor is to Unity as the D/W-
d^nd IS to the ^otient ; arid fo the Diftance between
Unity and the Divifor is equal to that: between the
Dividend and patient. If then the Fradion ef be
divided by ab, becaufe aA is :sec, therefore cd is the
^uotienty the Index of whofe Logarithm is tc ; but -
tc=t A-He — ta. Alfo if the Integer CD be divided
by the FraSlim ab, becaufe a A=CE, therefore EF
i^ the integral ^oiient whofe Logarithm is tE ; but
tE=tA4-tC*^ta. Again, let th^FraBion ab be di*
Vided by. the Integer CD, becaufe CA=ac, there*
fore thq; FraSiion cd is the ^otient \ whofe Loga-
rithm tc=:tA-|rta — ^tC \ that is, in every Cafe, the
Logarithm ef the Divifor being taken from the Loga-
rithm of the Dividend^ if to (he Remainder, yoa add
the Logarithm of Uniity^ the Sum will be the Loga-
rithm of the ^otient : which is but the Converfe of
the RttUs for the Logarithms in Multiplication^ as is
evident by Infpedtion of Art. 13. foregoing. And
the Methods there mentioned for duely adjuiling the
Indices^ are to be equally obferved here.
. i8. In Involution^ or raifing the Powers oi Frac*
tionsj 'tis evident that theDiJiance between t7»//y and
the Root^ is equal to the Diftance between the Root.
and thejf/y? Power ^ thtfirft gindfecond Power j the
fecond find third Power^ and fo on. Therefore fince
Aa=ab=ce=aeg^ 6fr. cdzz: Square ofab, efzizCube^
gh= Biquadratej &c. Power of the Root ab. But
fince AB : ab :: ab : cd :: cd : ef :: ef : gh, (^c. there-
fore the Logarithms t A+tc=2ta, and fo tc=:2ta — ^t A.
That is, the Logarithm of the Square of the Root is
equal to double the Logarithm of the Root lefs the-
Logarithm of C/»/7y. Again, lit!? =atc«2ta— tA,
and fo ta4-tc=4ta — 2tA ; that is, tex=:3ta — 2tA 5
in Words, the Logarithm of the Cube is equal to
triple the Logarithm of the Root^ lefs double the Lo-
F garitboi
34 TieNaturetf Log AJLiTHMs, and
garithm oiUnUy. Moreover, ^^i^ t9=3ta — 2tAt
therefore tc + tg=6ta — r4tA ; but tc=52ta — ^tA,
fubduA this from the kfi EquatioHj there will re*
main tg«s4ta — jta i or the Logarithm of the Bi- -
piadrate.
19. And uMiverfalfyj if the Pinver of a Frac^
tion be x^ and the Logarithm =: L9. the L <^a-
rithm of the Power x (hall be esxL — ^*'-— i xtA,
or ass ^ L— X t A 4- 1 A. Thus, fuppofe you would
know the JLogarithm of the Square of the FraHim
dc ^ here x=r2, and Lsstc, therefore 2tC— tA=tg»
the Logarithm of the Square gh, as required. If the
Logarithm of the Cube bedeftted of the FraSltontf^
we have x* ■= 3, Lrs te, and fo jtft— 2tA=tr, the
Logarithm of the Cube rs, as deftPd. And thus you
proceed for the Logarithms of other Pozvers.
N. B. The Indices of Logarithms of all Paweri
of Fr off tonal Numbers (l mean (iich as are furelf
k>) muft be doubly dotted^ finice thofe Powers al-
ways fall below the Roofj which is fuppoied a
pure FraHion.
20. Evolution^ or the ExtraBion of the Roots oS
Powers^ is juft the reverfe of the foregoing Procefs.
P or fuppofe^ the FraSiion cd were given, whofe fauare
Root was required* Becaufe AB : ab rr ab :^, thete-
fore AB x cdssab xab ^ and fo^ AB x cd=ab ; there-
fore the Logarithm — "^-^^ =ta, is the Logarithm of
ab tht fquare Root fought. Alfo if the Cube Root of
th^Fratlion ef be required ; becaufe AB : ab :: cd : efj
we have AB xef=ab xcd ; and fo the Logarithms
tA + tc=s ta^-tc. But tA+tc— ta=5:tcs=2ta — ^tA,
that is, 3ta=2tA — te, and confequently -^^--iS rrsta
the Logarithm pf ab the Cube Root required.
2fr
Chap.IV. their ImUceSy farther expUtined* 35
21. And univer&lly, if the Logarithm L of the
Root of any Ptmer x, of any. Fra^on rs, be requiied*
we have this«.<>rm tr+xtA-tA ^^ ti+f=mA
=L=ste, that is, in Words, the Number* — ixtA
added to the //ri^Ar of the Logarithm of the Fraiiian^
the Logarithm thus augmented being divided by x^
the Siuotient /hall be the Logarithm of the Root
fought. Or, iincetAe=io, lOO, the Number at — i
preftx'd to the Index of the Logarithm of the Power,
and the Lo^rithm thus divided by x^^ the Sluotieni
ihali be the Logarithm of the Root fought.
C H A P. V.
^tbe original ConJlruSHbn of Logarithms by the
Lord t^cpcr^ and the Alteration thereof to the
prefent Form by himjelf and Mr. Briggs, ex^
plairidandilluftrated by /i&^ Logarithmic
Curve.
I. riri H E noble Inventer of Z.e»/irr//£«r/, the Lord
X Neper^ having duely contemplated their
wonderful Nature, firft con(uruAed and publiihM a
Canon thereof; but thofe far different from what we
now comnKMily u(e. And this was no Wonder, fince
Scarcely any thine receives its Invention^ and utmoft
PerfeBion at the feme time.
2. In the firft kind of Logitfithms that Neper pub-
liflicd, the firft Term of the continual Proportionals^
was placed only fofar diftant from Unity/ tis that
TVnw exceeded Unity. Thus, for Example, if o n
be the firft Term ox the Series from Unity AB (fee
Fig. I.) the Logarithm thereof, or the Diftance A n,
or B y^ was by iiim put egual to vy or the Increment
F 2 of
5$ ConJlruStion ^Logarithms,
jof the Number n y above Unity. If then we fuppoA
yn=3H .000000 1, the Excefs of this Number above
•AB=:r, iso.ooooooi, which therefore, by him,
Yf^rmdtit&Logarilbm ; that is, An=0.ooooooi.
3. From hence, by Computation, t}ie Number
10 will be the 23025850th Term of the Series ; which
Number thej-efore 1^ the Logarithmoi ip in this Form
oi Logarithpts ; ^nd expreffis its Diiftancc from Unity
ixi fuch Part whereof vy, orAn-i^one. Alfo the
Logarithjn of 2 (in this Form) is 693 147 1 ; of 3, is
10986122; of 4, is 13862943, {i?r.
4. But this Pqfition of the Ratio of the Terms Is
entirely at Pleafure \ for the Dijiance of the /r/? Term
may have any given Ratio to its Excefs above Unity ;
that is, A n may be indilfferently lefs^ equal to^ or
greater than vy ; and according to that various Rath
(which may be fuppofcd at pleafure) between A n, or
By and vy, /. e. the Increment of thcfrft Term g-
bove tlnity^ and the Drftance of tht (eimt from Unity ^
there will be produced different Forms of Loga-
rithms,
5. The Logarithms of this Jirji Form^ were found
by thcfagacicus Inventer not to anfwer the Defign in
the beft manner as could be wi(h*d ; and thereforb
he changed *em into another more convenient Form^
wherein he putthe Number 10, not asthe23025850th
Term of the Scries, but the locoooooth Term : And
after Nepefs Death, the learned and indefatigable
Mr. BriggSj with great pains^ made and publiih'd a
Canon of Xiogarithms according to this new Form.
Now fince in this Canon the Logarithm of 10 is
1. 0000000, and fince i, 10, 100, 1000, £s?f. are
Proportionals^ they fhall be equidijiant from each
other ; wherefore ' the logarithm of 100 ihall be
2.0090000 ; of 1000, 3.0000000, and the Loga-
rithm of 10600 will be 4.0000000, and fo on. And
this Form of Logarithms hath been ever fince in ufe,
pd are thofe in pfefpnt IJfe ; the Nature and
Properties
Chap. V^ ipeplained and illujirated. 37
Properties of which we have been hitherto explain-
ing.
6, The Rationale of the Method by which Mr.
Briggs computed his Logarithms, is beft explained
from the Lt^aritbtnetic Curve^ according to Dr. Keil^
as follows. In the Logaritbmetic Curve HBD (Fig. V.)
let there be three Proportionals AB, ab^ qs^ very
nearjy equal to one another \ that is, let their Diffe*
rences have a very fmall Ratio to the laid Ordihaies^
(for fuch are thofe Proportionab,) and then tht Dif-
ferences of the Logaritoms will be proportional to the
Differences of the Ordinates ; that is, it will be
sr : be :: Br : Be :: Ag : A<s, For fince the Ordinates
AB, ab^ qsy are nearly equal to one another, they
will be very nigh to one another ; and fo the Parts of
the Curve Bj, B^, intercepted between them, will
nearly coincide with z-Rigbt Line ; for it is poflibic
that the Ordinates may be fo near to each other, that
the Difference between the Part of the Curve and the
Right Une fubtcnding it, may have to that Subpenfe^
2l Ratio lefs than any given Ratio, Conleouently the
Triangles Bcb, Qrs, may be takea for Right-lined
ones, and will ht equiangular : and therefore, fince
a b is parallel to qs, they will htfimilar^ and their
homologous Sides proportional j viz, rs : be :; Br : B^,
or Kq : Aa.
7. From hence, by the way, appears alfo the iS^^-
fon of the Correction of Numbers and L^aritbms by
Differences and proportional Parts. For putting
AB=i, or Unity, 'tis evident, that the Logarithms
Be, Br, are proportional to the Differences cb^ rj, ^of
the Numbers AB, ab^ rs ; as we Ihall hereafter prove
by Fa£ts^ in the praSical Part.
8. If a mean Proportional ht found between I and
ID, and then again another Mean between that and
Unity ; and if proceeding thus, you continually find
a mean Proportional between the Mean laji found and
Unity ^ bifeifting the Logarithms ftill as you proceed
(in
38 QmfiruSiion ffViOQK'Bi i t h m s,
(in the manner of the Ewmpk in Ch^p. 11. Art. 6.)
you will at laft get a Number whofe Bijiance from
Unity ftiall be Icfs than the tooooooo-ooooooTo Part of
the Logarithm of 10.
9. After Mr. Briggs^ in this manner, had made
54 Extraaicns of the Square Root^ he arrived to the
Number i. 00000 00000 00000 12781 9149320032-
3442, and its Logarithm was 0.00000 0000000000-
^555^ 1 1512 31257 82702. Suppofe this Logarithm
be equal to Aj or Br •, and let the Number found by
thbExtradion, be=:;i; and then its Excefs above U-
nity will bc=rj.
fAj=aO.OOOOO 0000006000 0555 1 11512-
1 3125782702.
ThatiSy^ ji =1.0000000000 00000 1 2 78 1 91493-
20032 3442.
rj=so.oooooooooaooooo 12781 91493-
200323442.
low by means of tbefe Numbers the Loga-
rithms of all other Numbers maybe found in the fol-
lowing Manner. Between tht given Number (whofe
Logarithm is to be found) and IJnity^ find (by the
ExtraAion of Roots, as above) fo many mean Pr9^
fwrtumals till atlaft a Number be obtain*d7& little ex-
eeedingUnity^ that there be 15 Cyphers next after it,
and as mznyjignificant Figures after tboie. Suppofe
the fmall Number thus found h^ab^ and let the^^-
nificant Figures with the 15 Cyphers prefixed, before
them, denote the DiiFcrence be ; then fay, as the
Difference rs is to the Difference bcy fo is the given
L$gariibm Br, to Be the Logarithm fought for the
Number a b. If now tRis Logarithm B^ or Aa^ be
coniinually doubled the fame niimber of times ^ there
yreft Extra&ions of the Square Root^ you'll have at
]af( the Logarithm of the Number propos'd as re-
quired.
II. If the Tangent TB be drawn to touch the
' Curve in B, thcii may the Subtangent AT be found
by
10. N
Chap. V. explained and ilbijirated. 3^
by: the Numbers above in jlrt..^. For fince AB,
Br, are parallel torj, AT -, therefore the RigU4hfd
triangles Bri and BAT, are fimilar^ and fo as
sr : rB :: AB ; AT, theSubtangent j but fince ABsi,
therefore ^=AT ; or thus 5
sr
As rx =0.00000 00000 00000 1 2 78 1 9 149} 10032-
Is to Brzso.ooooooooQO 000000555 1 11512 3x257-
8270,
^ Is AB=::1 .00000 00000 00000 00000 00000 OOQOO
OOOO,
to AT=o.434i9448 190325182^65 1 1289 18915.
. 6051.
12. If the proportional Right Jams GH, EF, AB»
CD, (Fig. V.) aie Ordinate to the Axis CV of the
Logaritbmetic Curve^ and if their Ends FH, DB, be
joined by Right {Jnes^ which produced meet the
Jxfs in the Points P and K, then the Rieht Lines*
GP» KA, will be always equal. For fince GH : EF
::AB:CDi itwiUbe, asGH:Ft:: AB : DR (by
Chap. II. Art. 8.) But becaufe of the Jimilar Tri-
angles PGH HtF, as alfo KAB BRD, we have
PG : Ht (:: GH : Ft :: AB : DR) aKA : BR.
But HtsBR, and therefore PGsAK ; which was
to be denHMt^rated.
13. If the Right Lines CD, EF epially accede to
AB, GH, fo that the Pbint D at laft may coincide
with B, and the P<Mnt F with H, then the Rights
Lines DBK, FHP» which did hfore cut the Curve,
will now only touch it in the Points B and H \ that
is, they will be changM into the Tangents BT, and
' HV. And fothc Right-Lines AT, GV, wiUtfA
ways be equal to each other ; and fo the Suhtangent
AT, oa GV, in whatever Part of the Axis it be, is
always me cmjlant given Length ; and this is one of
the moft remarkable and ufeful Properties of theZjGK
garithmic
40 A Method of conjlruSiifi^
garitbmic Curve : For the different Species 6r Formi
of thofe Curves are deter m i hed by the SubtaHgents.
14. If the Excefs cb of any Number a b ejttremely
near Unity, or but a fmall matter txceedihg \t\ be
given, the Logarithm of its Diftance from Unity Aa
or Br, will be known by means of the conftant Sub^
tangent AT ; for by Art. 1 1. we have be : Be :i Afi
: AT ; therefore ATx^f=Bcx AB=:Bc, the Loga-
rithm required. Thus alfo AT x rj=Br, the Lo^-
rithm of the Number qr ; and fo the Logarithm of
any prime Number 2, 3, 7, 11, 13, 6fr. may be
found independently of the Logarithm of any other
Number..
^"^n^
C H A p. VL
ji Method of conJiruSiing the Logarithms derived
arni detnonftratedfrom the Nature of Numbers
only^ by Dr. Edm. Hallcy,
I. 'T^ H E admirable Method now before us, is
X one of the many great and wonderful In-'
ventions and Difcoveries of the celebrated Dr. Halley^
the prcfent Afironomer Royal^ and Fellow of the Royal
Society*^ and whofe Name amongft the Literati will
be had in ever lofting Remembrance. This Method
not only comprehends all the Improvements made
by others by means of the Hyperbola and other Geo-
metrical Figures^ but fhews with great Accuracy
from the. pure Properties of Numbers (as moft natu-
ral and agreeable in a Bujinefs purely Arithmetical^ as
the Logarithmotecbny is) bow the Lc^rithms may
be produced to any dejired Number of Places^ with
^r greater Eafe and Expedition than by any Method
before known. According to hio^ therefore^
2.
Chap.VL 5/*LoGARiTHM8. 41
. 2. Suppofe an infinite Number of equal Ratios or
RatiuncuU between any two Terms in a continued
Seale or &eries of Proportionals ; and thofc Ratiun^
cula exprefs the Ratio of thpfe two Terms, as of
I to i-Hx. If then between Unity ( i ) and any Num^
her propofed (i-4-Ar) there be taken any Infinity {n)o(
' mean Proportionals^ the ihfinitely Jittle Augment or
Decrement of th^firft of tbefe Means from Unity will
be a Ratiuncula or Fluxion (*) of the Ratio of C/zri/y
to the iaid Number; and the Terms of the- Series
will ftand thus ; viz. i. 1+*^. 1-+-^. i-¥x, i-hx
&c, to i-hx . and the'
Exponents} ^- ^' ^: 3; 4; g^- to «; or thus.
From whence *tis evident that not only the Number
C»j of the Proportionals or Ratiuncula^ but alfo their
Sum (nx) may be put for the Logarithm of i-f-^.
And thus alfoJV^ maybe put = Logarithm of i-hy,
and confequently it will be, as L, i+x : L, I-+7
:: nx : W,
3. If;e=i, that IS, if the RatiuncuU compoiing
divers Ratio's have the fame Magnitude, f hen are
thofe Ratio's proportional to the Numbers of Ra^
tiuncuU cbntain*d between their Terms, viz.
L, 1+^ : L, i^ :: n : JV, For if Af=i, 2Lndy=2 \
then i-4-^fc=2, and i-+y=3 ; and if »=3p|03, t?r.
iVwill be found =47712, fcff. that is, if there be
the infinite Number 30103, &?f. of RatiuncuU in the
Ratio of I to 2 ; there fhall be the infinite Number
/^yyii^ fcfr. oH)\Q hm^ Ratiunci4^ \vit\i^ Ratio oi
I to 3.
4. On the contrary, \f nis^N^ then L, i-^rx j L,
i+y :: x :y^ that is, fuppofe the infinite Number qf
Ratiunculae in one Ratio equal to the infinite Number
of RatiuncuU in any other Ratio^ then arc the Lo-
garithms of thofe Ratio's diredlly as their Fluxions^
G * or
42 A Method of conjirulfif^
or as the ^Magnitudes of the RaUuncuU refpedively.
For'inftance, let the Ratio of lo to |, jog to i,
*iooo to I, i^c. all and every of them be fuppofed
to confift of 5 Rat\unculae i as follows; Thus,
{Ratio's o. I, a. 3. 4. 5. &ff»
Terms i. 10. 100. lOOo. loooo. looooo, &ff.
Ratiunc. 51;. 5w. 5X. 57. 52, i^c.
*Ti3 plain the Ratiuncula v of the whole Ratio of 10
to I IS T 5 of 100 to I, is I, f5?r. that is, the Ra-^
tiuneuU are/i;=rj, w=t, ^rs^, j^=:t, 2rrii butthofe
Fr anions are as the natural Numbers i, 2, 3, 4, 5.
Wherefore the Logarithms of the Ratio's of lo to i,
100 to I, 1000 to I, Cf?f. are' diredly as the natu-
ral Nunibers I J a, 3, 4, 5, fi?*r..and fo may be ex-
preffed by them.
5. Since then the Logarithms of Ratio's are as
their Fluxions^ therefore the Logarithm of any Ntim* ^
ber is found by taking the Difference of Unity and ^he
infinite Root of that Number ; that is, becaufe 1+^
is the firft Term from Unity, ot Ratiuncula j i-^^"
:;=i4-Ar, is thei infinite Po^er to be refolv'd \ and
•7+3, or i+J''=?=|-4-ir, andfo i-f-^^" — i=3c^ Lo-
garithm of ^-+-Jif.
6. In order to extraft the Root of the infinite Power
iH-x, (which, to fome, may fcem ftrange and next
to impoffible) we muft make ufe of Sr Jfaa,c Newtqn%
celebrated Theorem for that purpofe* Suppofing
then the Power be iH-at, according to his' Theo?
rem, I -f- *"==! I + J^AP 4- -^JIT- ^Af + — 6^3 — *
,-6»4.,„»_6»» ^^^ ^^ ^^ ^^ ^f^^ p^^gp
i-^x^ when the Index (n) is finite ; but (n) being in
. the prefent Cafe infinite^ all the Terms of the Co*
efficients^ wherein (nn) is found a.Divifor (as being
inf^nitely in^uite) will vftniftij^ as being infinitely le^
chap. VI. e/'LoOAkiTHMs. 43.
than nothing. But the Co-egicient ~ = i-— 1 =-
Coefficient '-6>.+..;»-6>.f = ±^_ ± =: Ij &c.
Wherefore the foregoing Root will become i-^-i^ —
inx'''+ \nx^ — inx^-^-inx^ &c. That is^
t
ixx-^^lxx-hixxff — i*^ — ix% &c. =i+^* — l=si>=:
Logarithm of i '^x.
y. And whereas the infinite Index (») may betaken
at pleafure, ah Infinity of different Scales of Loga-
rithma may be produced ; and thofe different Loga*
rithms will be to one another as j^, or reciprocally as
the Indices (»). And as it hath been (hewn (in
.Chap, v.) that in making thejfr^ kind of Logarithms
by Neper ^ the infinite Index of the Logarithm of lo
would be 23025850, &?f. But in making the fe-
cond fort after by BriggSj the faid infinite Index was
put =10000000, Csfr. Confecjuently in the fore-
going Series for Logarithms, if «=iooooooo, dfc.
the Lord Neper^s Logarithms will be produced i and
the Scries will hejimpfy x^^x^^ix^—ix^J^ix*, &c.
On the contrary, if »=:23025850, &fr. thenBriggs^s-
Logarithms will arife from the Series j and bccaufe
» = >.3oa,8o f £sf^. =0.434229448, i^c. therefore
J(=:AT) is the Subtangent of the Logarithmic Curve
for the Briggian Logarithms^ as is plain from Chap. I.
V. Art. II. Whence if a Logarithm of Neper*s
Form be multiplied by 0.43429448, 6f^. or divided
by 2.3025850, ^c. it is converted into a Logarithm
or Briggs's Sort^ or thofe in prefent Ufe.
8. If the Logarithm of a dccreafing Ratio be
fought, as of I toi— X, the Power being i — x^ its
■L
infinite Root will be i— x*=i— i;^— *»^*— i»*'—
G 2 iwx-*—
44 A Method of conJlruBing
inx^—inx\ &c. that is, i x x-f ix^-H^^+ir^+TJf *»
&c. =1 — I — :e^z=Sif=i the Logarithm of i — x j the
firft "Term next below Unity, or Root of this infinite
dccrcafing Series being i— ;r. And fo in this Cafe,
according as the infinitelndex {n)is made =10000000,
6fr, or 2.3025850, 6?r. fo Nepet^s or Briggfs Loga-
rithmsof thofe negative Numbers will be produced.
9. Inftead of the Terms i : 1+;^, let ^ : * expreis
the Terms of any Ratio univeriaily 5 and make
^+^=^» ^^d a — b=::d ; and fince it is i : i+x 11 aih^
therefore /i+^*!=i, andfox= tzi — L . Again
becaufc (in the decreafing Ratio) it is i : i— jc :: ^ : tf ;
therefore b—bx=a^ and fo we have again x = t^
= 7 • Whence the Logarithm of the feme Ratio
a : b^ may be doubly exprefs'd ; viz. for the encreafmg
Ratioj the Series will be
the decreafing Ratio 5 all which is evident from the
. three laft Articles.
10. But if we fuppofe the Ratio of a to b^ viz. -^
compofed of two Parts ; viz. of the Ratio of a to
the Arithmetical Mean between the Terms a ^nd ^,
and of the Ratio of the faid Arithmetical Mean to
the other Term by that is, fuppofe ~ =-^ v ^
(for is = tt^ is the Arithmetical Mean between a
2
and b'^) then the Sum of the Logarithms of thofe two
Ratio's, ~> ^, will be equal to the Logarithm of
^e Ratio of a to b. Or, L, ^ + L, ^ =L, ^ .
And
Chap. VI. g/'LpGARITHMS/ 45
And thus alfo we have L, ii+Li^Li
Now bccaufe the Ratio of 4i to * is encreafing, thctc^
fore I :i + x::is:l^i and fo i -f ^= i; con-
fequentlyx= •; i= -y-^ = — a-new; again,
becaufe the Ratio of is to a is decreajing ; therefore
4j : ^ :: i : i — x, and fo i — ;^ aes J! ; and
1 1. Therefore fince ;f = i for both Ratio's, vi%.
of tf to ij, and *j to ^, we (hall have (by the fore-
going Rule, Art. 9.) Jx 1 4. 4 4.-I!! 4,^
4--^. 6?r. =(A)L,tfto4j, andjx i_j:
+ 3? ~;^*+P'^^- = (»)L.4,to*. Then
.-'' 7 • + ^ • + ~, .t?^. = (A+B)L,ato*.
Thus you have a Series exprcfling the Logarithm of
the Ratio of ^ to b^ whofc Sum is szzaJ^b^ and Dif-
ference d=<j4"^' and this Series convenes twice as
fwift as the former in Art. 8, and therefore is more
proper for making or ^flf^w/W^g- of Logarithms, which
it performs with great Expedition, when d the Dif-
ference is but the looth Part of s the Sum ; the firft
Step — fufficing for 7 Places of the Logarithm, and
thefecond -7 for 12 Places.
12. Becaufe the Difference of the Logarithms of
the Rations of ^ to is, and is .to ^ is the Logarithm
of the Ratio of a ^ to |ss ; or thus, becaufe
^)
46 A Method of confiruSting
(A-B) = J X ^'^. -f ^ ,^4.. 1^- , ' 6fc. but
half the Ratio .-is the Ratio ^ (for J^ x ll? -
^), that is, the Ratio of the Geometrical Mean
y^l^b to the Arithmetical Mean i s j confequently
theLogarithmofV:g= J x i! . ^^t . -f Ji;,
&? f , which IS a Theorem of good Difpatch for find-
ing the Logarithm of is.
1 3* But the fame Logarithm may yet be much more
advantageoufly obtained by a Method like the for-
mer. For if we make the Terms of the Ratio
^ =-^, and put 5 = tfi 4. iss, the Sum of the
Terms , and D = ^ ^ — iss ; 'tis evident the
i2l » f2!, ^c. by Art. 11. But becaufe iss ='
ia^ + iab -f- i^*f therefore D = tf i — iss s: tf ^ —
id" -{-iab^.ibb =« 4^* — iab + i^^ = v^l^^^TIJ
= id* =1, in the prefent Cafe of finding the Loga-
rithms of Prime Numbers ; for fuppofe the Loga-
rithm of 23 be fought, then ^==22, ^= 24, issziaj,
and d 3= 2 ; alfo -3^=^^ = 528, and fl=isss=s529,
and therefore ab — hs;=:A—Bz=Dz=i(^:zll. Where-
•
fore, finceD=i, the Series above becomes ^y, — j.
■iS5 + j-f^ + tT^' ^'- ='^^ Logarithm of the
Ratio of -r* i a»d fo the half of it, viz. ' the Series
i '^ i + pi + iH-s + ?5^' ^^- ■ =*^« Loga.
richm
Chap. VI. g/'LoGARiTHMs. 47
rithm of the Ratio oi^ab to is. • And this converges
much fooner than any Theorem hitherto invented,
and beyond which nothing better can be hoped for,
in tht great Author's Opinion.
14. The Logarithm given to find what Ratio it
expreffes, is a Problem lblv*d with like Eafe^ and de-
xnonftrated by a like Procefs to that foregoing for
finding the Logarithm of ^ given Ratio. For as the
Logarithm of the Ratio of i to i-j-^ was proved to
be i^" — I , (by Art. 5.) and that of the Ratio of
I to I— ^ to be I — I — x» rby Art. 8.) fo the Loga-
rithm, which we will call L, being given, fincc
L=i+;tf» — I, therefore L+i=i4-^ in the firft
Gafe; and i — L=i — »vw, in the latter: Confe-
quently i+L =i4-^> fl'id i — L =1 — x. That is,
according to Sir Ifaac^s Theorem^ l-|-»L+»^*L.*+
J»«L«-f ^♦L^+TT^w'LS £s?c. r=i-|-;ri and alfo
I— 11L4- 4ii*L*— . iri^U-ir ^n^l-^— ^izn^L^ &?r.
x=i — X ; confequently the Number i+x or i — x
is readily known by thofe Series, be the Species of
the Logarithm what it will. That is, whether it be
iVif^^r's Logarithm, where »=t=iooooo, &c. and fd
l±xz^l±L+iU±lU+^\L^±^i^L^, i^c. orwhe,
tlier »=2 3025850, &c. for Briggs^s Logarithm.
15. If one Term of the Ratio^ whereof L is the
Lx)garithm, be given, the other Term will eafily be
had. by the fame Rule. Let a = the leaji Term of
the RatiOy and b = the greatefk •, then becaufe it is
I : i-i-xiia :ti and foi-f-xss — =ri^L +
. iL* + IL* + T^L^ ^c. if»=ioopooo, &c. and
therefore b=s: a +'tf L + i<^L* + iah\ &c. if a were
|iyen j but if ^ were given, becaufe i ; i—x :: h a^
therefprQ
4$ A MetbodofcorjftruBing
therefore i — y = 4 » *^d ^^ a^=zb — ^L +
i^L— i^L*+ AL, esfr. Wherefore by the Help
of the Tables^ the Number belonging to any Loga-
rithm will htexaSly bad to the utmoft Extent of
the Tables.
1 6. Suppofc — = -A^f the Number belonging to
the given Logarithm L of the Ratio — , then
1 == tf, let the Logarithm of the R^tio ^ be 3,
and let the Term b b^ known ; then (per Rule,
Art. 15.) we have i : i—x :: b : JNT, and fo i — xi ==
i?, andiVs:* — ^ + i*aa— l^asfcf^. if a be
Nepey*sLog2LTithmyhutN=.b — bn'^+ibn^'i'B — ln*h^y
6fr. if 1; =5=2. 3025, &c. as in Briggs^s Logarithm.
But if the Ratio be 4 == JV, then ^^ =; *, and fo
I + ^ = ^ , therefore iVsatf+tfa+itfaa+I^S 6?f .
Or N.= a^an'3i+ ian'"h*-^an^'^\ iSc. Note, here
a and ^ denote the Number belonging to the neareft
ntxt lejfer ornext greater Logarithm than the^/i;^»
Logarithm L, and the Logarithm g is the Differenee
of thofe Logarithms ; wherefore as a is I'efs, the Se-
ries converges thefwifter ; and finds the Number N
of the Logarithm L, much, iboner and eafier than
the Rule in Art. 15.
1 7. In the foregoing Series tf+^H-f i^aa+i^S^?^.
x=N\ the three firft Steps may be abridged thus,
a -j J a = ^ + ^a -|^ 1^33, very nearly, and may
ferve with ExdSnefs enough for Numbers not
exceeding 14 Places, which is more than fuffi-
cient for common Ufe. Therefore we may take
a^
3
1
trhap. Vi. tf t od AHt T Hli«; 41
a+ l^^N, or3 — ^=:i\rjand if the
Jfei/«r » taken for Bri^h Logarlthihs, wc A^ have
^^T^^^ N,ov b^ l£^.=^Ni that is
**rl^=i iNTj which Ecjuation may be rcfoIv*d jnto
the following Analogies ;
w;^. i ^~!^ • "^"^3 •• ? i ^5 Ofi
€8. If more Step^ pf /i^f 8mu be defir^d, it wU)
be found as foUows, i^/x ^^^ fJL- — "dfi* -x,
^i^ ,&c.=!^Ari alfb theiRule t-fwL+iMLL^*'
i»^L', 6ff. may be thus contracted, Hz. i -^^
2Hh»I>H»»LL xi^Lss JV. What is laid concern -
ing tKs Method 'of making, ^rither LigaHtbms or
Numb^s^ J ()fic6iGi^;ls fuftcient to ^dcrit very in-
telligible to any common Capacity; and to ihew the
admirable UJefulntfs and Exc^lkniy thciwf bi^Md
iny other^ Mtherto ifivented.
1^ r CHAP;
]^d^ 72^ Logarithmic Series
CHAR vn.
1t^ * Logarithmic Series aforegoing^ demons
firated aljb ^ *Fluxions, amfroni the Nth^
' ture of the ♦Hyperbola.
x.rr^HE preceding Series for the Logarithms,
X ^bich has, been demonftrated furelj from
jdrithmeticd Principles^ or the Properties of NuM"
bers^ may a!(b be prov'd from the Do6bine of the
fluxions of L^ariibms. For the Writers on Fiux*
ions varioudy aemonftrate she Fluxion of the Loga-
rithm ofas^ Number is equal to the Fluxion of that
Number, (wbofe Logarithm it is) divided by tbefaii
Number it f^.
2. Let the Number propofed be i+^, the Flux-
ion of which is jr, therefore the Fluxion of its Loga-
rithm will be ss -~- 5 from whence the foregoing
Jnfioite Series for the Logarithm of the Number i+^
may be derivedi as fqllows. The Logarithm <^
l-^x is equal to the flowing Quantity or Fluent of
the faid Fluxion -i^. But ^r"^ ^ x -j- • and
Jj^vszl+X) I (=1— X-^2J*P— XS &C,
i-^x
• — X
— X — XX
'+XX
'I XX \ x^
»
-X*
-tVf*, &c.
chap. VII. JeHmftratedby Fluxions. $t
3. The Quotient then i—H-x*- ■ ft* I » S Uc.
1_ J but *x 1— *4-x*— «»+x*, &c SB i~
j^jj4.^,ji«_,c»i-f-J^*,&c«— -^. The i7««i/ there-
fore of that infinite fiuxienary Sam (by the inverfe
Method of Fluxions) is found to be *L_i^^4^_
»^_i_|x', &c which therefore is the Logarithm of
dje Number l+* ; and is the very fame with that
Scries in the foregoing Chap. VI. Article 7. for Ne-
pet*s tfigoritbms.
4, Again, if the Ratio be decreqfing, or the
Number be 1—*, the Fluxion of this alto is*, and
therefore the Fluxion of its Logarithm ^. But
i__ « ir X -i- as lH-;f+X»+*», &C.X * ss A +
l—x «— *
xi-f ^*+x»x, &c. The Fluent of which is *4.fe(»4.
7X»4-i**, &c. the fame Series as that in Article 8. of
the preceding Chapter^ for Neper's Log^uithm of
5. The fame Series is likcwifc deduced from the
Nature or Equation of the BfperMa. For letFCH
be an Hyperbola (Fie. VI,) AE, Al, the Afymf totes 3
draw BC, DC parJlel to AI and AE j alio draw the
Ordinate EF parallel to the Ordinate BC, or Afymp-
tote AL Let AB=«, EF==y, and BEsswf. The
Equation afl=tfy-f ;gf expreffes the Nature oftheiify-
perbola between tU Afymptotes. Now the Fluxion
of the Space between tht Jifiiffaf Ordinate^ and
Curve, is always equal to the Proek^ Qi the Ordi.:
nate into the Flumon of the Abfciffa j that is, in this
Cafe, s»ri. Therefore to determine the Fluxion of
the Afymptotic Space contajnM between the Abfcifla
BE, the Ordinatea EF, EC, and the Curve of the
HjrperbolaFC, we have ^-^^s J', thcK^fore^ as
f *= Fluxion of the (aid Space FCBE.
H 2 6.
'^4 *thi ZAgariibrntf Smtt, Bcq>'
6. But ^ =r ^ xi, and ^ = a-f*) a0
(=«— *-f.Sf_i!H- '4, &c? Wherefore ^ =
MTf ^antity of this fluxionary Series is ax — : be* -¥■
fa-T^-^^^ *^' - ^^ Space FCfiE. Sup.
pofe^j=i, then x— |yM-4^^— i^^-Hx% &c. = tbc
{kidyij^fnpiofic Space FCBE, as before, But 'tis cvi-
4ent this is again the very Tame Series as was invented
by Pr. Halley for the Logarithm of the Number i4^.
7. The ^ymptotie Spaces^ then, are with rcfpcft
to the Abfcijfie^ as Logarithms in refp^ to Numbers.
Thjit is, Cnce AB=i, and the Logarithm of 1 is=o^
the Spaces Bu^C, BfiC, B^^G, Bi&)fcC, 6?r.
are the Logarithms of the Numbers A tf, A r, A ^,
A i5, &c. Again, be^aufe the Jbfcijfie are in a r^-
tiprocal Proportion of the Ordinates^ that is,
AB : AE :: EF r ,BC •, thereforethe Afymptotic Spaces
are in refpeft of the Ordinates as Logarithms in re-
foeA of Numbers: Yet fo, that while the Ordinates
BC, ab^ cd^ eg^ hk^ EF, iecreafe in a Geometrical
Rirtio.tht S^caBabCj BcdC^ 6?r. may in-
cfcafe in an Arithmetical Ratio. And fince in Ncr
per*s Logarithms J x x --^ixx'+^^xx-^ix^y &c,
praiopooQt &Cp *tis plain his Logarithms become
the fame witK the Hyperbolic Logarithms juft now
^pnfidcr'4. ' '
Sf
CHAP,
Chap. V|II. OfhoGAKnnMSy&c. 13
C H A P. VIII.
fTbe Method of conjlru0ing Logarithms by the
Infinite Series^ exemplified and iUuJlrated.
l.fTPiHE Manner of raifing Theorems for the
X Conftrudtion of Lc^rkhms hath been luf-
ficiently explained \ it theretore remains that we il-
iuftrate the fame by proper Examples. TkeTheo^
rems for doing this direftly are.
Theorem I. J x at =p *;? *-4- 4x'=p ixH- \x\ &c.
-s L, izbv.
Thtoftmir. J X f 4- ;-^ + ^^ + ^\ 6?f.
— T *
Theoremlll. i x ^i + ^ ,-+- ~ 4- ~,
Theorem IV. Jx|- 4- jgj -*-j-p -♦- -g,-
&c. = L
.Note, in thefe Theorems, J is all along applied to
^apt them to dl forts of Logarithms.
2. Since »=the Logarithm of lo, we muft there-
fore firft fupppfc »=iooooooo, &?f. and thence Ne.
per's Logarithmi will be produced j and fo thefe are
the firft fort of Logarithms which Nature affords:
TheothwB, as Brigg^s Logarithms, &c. are made
from them. In order then to find a Briggian Loga-
lithip, 'tis neceflaiy firft to find Utptrh Lo^rithn\
of iQ. • This may be done feveral Ways, either by
the Number lO itfelf, or by ife component Parts. If
we attempt it by the Number ip it felf, then becaufe
54 Of conJlruSUng Logarithms
i-hte:io, we ihall have x=:9, which, becaufe it is
greater than i, will occaiion that the fir ft Theorem
will not converge % and fo the fecond Theorem muft
be ufed. In this t-=tv> and therefore i= i + lo
ss 1 1, and ^io--'i=9. And thus the fecond Se-
ries for Neper's Logarithm of lo will be ^ H-
2.30258, &c. the Logarithm fought.
3. But this Series converging fo eiHremely Jlow^
renders the Bufinefs very tedious, and therefore the
faid Logarithm muft be attempted from the com-
fmtent Parts of lo. And fince 8x ii=:io, and
2 X 2 X 2 =?: 8, therefore 3L, 2 H- L, li = L, 10.
Confequently by finding the Logarithm of 2 and li,
we find the Logarithm of 10. Now Neper^s Loga-
rithm of 2, is found either by Theorem I. which con-
verges very flowly \ or by Theorem II. which con-
verges much fafter; and therefore to be chofen.
Here ^ t=z }.^ and /H-*=^=3, a — b=d^i^ and fo
the Theorem £? 4- fl' 4- 5^ , .fc?r. = ^i -t^
&c. = L, 2 ; this multiplied by 3, is = -^ -j-
3
ix ^ + ixf7.,£5?r. = 3L, 2. Nowput^,=A,
andbccaufc ^-r x p i= p , and' ^x = p therer
fore alfo put f A -B, and foB^z t. •, and thus
|B = C, 4C = D, and foon. Whence the The-
orem will become 2 -tr ? A + i BH- 1 C +| E) -H
Chap, VIIL hy infiniti Series. 55
4. By the fame Theorem IL we obtam Nefeft
Logarithm of li; for becaufe -i = li, therefore
L «,iy,(^i. = f.. Whicea+3=.i=9,
and d = tf '^^ =? I ; and fo the Theotem ^-f-
^%, ^c. =|^-txL + ixJJ. e?r.
2d3
But becaufe I
= iD=sE, (^c
therefore the (mcI Theorem will become AH- iC+
iEH- fG4- SI-I- T^L, 6ff. =L, 1 i as required. If
now this Series be added to the foregoing (m Art. 3.)
wc fhall have the Theorem 2+ itA+ 18+ ilC-j,
tt)+ ifE+ tVF+V1^4. %VH, fcff. = jL, 2+L,
]4=L, 10. See the Operation in the Table below. ' •
I.
I A=;0.22222222222Z2
B =: . . 246913580246
C = . . . 27434842249
D=. ...3048315805
E = .... .338701756
•'? = 37633529
■M.== 4181503
H= ....... 464612
I =;: . 51624
K=. ...:... .5763
L= 637
M= 71
2. :
llA:
tB-
*»C =
iD=s
I^E =
tVF =
tVHs:.
18 T
: 2.ooooooooooooe
:. .29629629629^2
:. . . 49382716049
:.. . 13064210595
: 33870»75^
; 9^P4io
: 2894887
876124
I... .2733Q
■- 8454
273
85
Thus Neper's Log. of io=2,3025850929940,&ff.
5. The Logarithm thus found (if continued on) will
be 2.30258509299404568401799 14546843642076
01101488628772976033328, &r. =»» and there-
fore
56 Of am/iruSting Logarithms^
fore in making the Briggian Logarithms^ we iKall
liave j^=o.43429448i90325i82765ii2'89i89i66o
50822943970058036665661 14454, &c. the Reci-
procal of the former ; which hencefcnth let be call'd
% ; that is, let J^. And now we are prepared to
iBnd the Logarithm (of any Form) of any oAcr
)«fiimber.
6. For Example, let it be required to find J?r/jjyA
X^Karithm of z, to 10 Places of Figures. In order
to mis, the Index J muft be aflTumM of a Figure or
pm more than the intended Number of Places ki the
Lo^rithm. The fecand Theorem is moft proper for
this Purpofei for here again i = f , is=: i, and
J S3 3, and alfo zso.434294481903 ; and the
Theorem Jxi? + !^ + ?!!, I6c. =zx| +
'}x^ + 1 — , ^c. =L, 2, and therefore -^ -f*
txp +ixp,&c. = JL,2. Hetcput^=:At
and becanie ^ s — x ^ s:txA=:B, and thvs
*. s«— X ^ =fBs=BC, andfoon. Therefore
S' 5^ 3*
the (aid Theorem will here again become A4- f B+
fC-f- il>-hiE-l- iSF, £sfr. = tL, 2, as is evident
:&om the following Operation.
I«d»
Chap. Via, fy hifimte Seriet,
$7
\z =UV. =0.144764827301
A— 0.144764827301
B= 1 60849808 1 1
»B — 5361660270
, C=s 1787220090
K= 357444018
D=s 1985800I0
ID- 28368572
Esa 22064445
jEsas 245*605
F = 2451605
t^F= 222873
G= 272400
AG=s 209^3
H=s 302^6
.Miss 2017
I = 3362
•M= m
K« Z7i
.-^Kaa 19
L=: 41I
AL=s 1
The Sum is *L, 2=0.150514997826
Moldi^edby 2
The Briggian Logarithm of 2=0.3010299956521
7. This Logarithm may yet be much eaficr and
fooner obtain^ by this Confideratioti, viz. That
1""= i^» and -JIHxi^^ =r5b* therefore
L ioee I T \ T «
r5i4^^^Tg7o ^s r* «o»t =3 L 4 = L 2. But
10
10
L||ff=:LiH* Whereforeput J|i=-J a-new, then
4i+i==^sss253, and^h— *s»i=3i thusTheofemll.
will convergsB much After, and will become, in
Numbers, z xTxTfl
fx
2533
«$3
Or if — =jr, the fiid Theorem, in Species^ h
j^ _f. |2jy» -f- :^zy, &c. Snppofe 225^ s= A, then
|z^»=Jx2ayxy*=3iAxy*=B, alio ^iy'=r|x2zyxy»xy»=s
iftcy*=C, and fo onj and thus the Theorem is
A-KAj'(=B)-MBj*(=:C>HQV(=D)+iiy,&c.
8.
58 OfconftriiSlingLoGAKiTKMS
{2zy=A= 0.010299473879 1 2
tAx5!X=B= . . 48271995
iBxjy =C=: . 4P7Z
TheSum is the Logarithm I ^^v^«^^^^/:/:«^q^
^f J_06o/ T 115X^1 *.. 0.01029995063980
Add the Logarithm of ~; . . . 3 .oooooooooooooo
And A of that Sum will 1 /:/• o
beL2== I ..0,30102999566398.0
Thus you fee 3 Steps of the Series thus ordered, are
fufficient for 14 Pkccs of Figures, whereas before
(Art. 6.) 1 1 Steps produced the Logarithm true only
to 10 Places.
9. Let thp next Example be to find the Briggian
Logarithm of 3. This may alfo be don6 by Theo-
rem II. where J^ = f-, and J=2, and j=4; alfo
~ rr i, 2=0,43429, &c. as before. Then 2 x ^ +
3S» T js' i 8 ' 32' '"•
= L3. The half thereof^ +7X^ +^x *
&c.= i L 3. Put -5 S3 A, then 4 = ^ X i =
Z o Z
^A = B, and ^^ = -^ xit:^i^ = C, andfoon.
Whence A 4. }B+ iC+ tD+ iE, &?r. =iL 3.
But this Series converges fq very flow, that as many
Steps will be neceflary as you intend Places of Fi^
gures in the Logarithm, and more ; therefore Re-
courfe muft be had to Theorem IV. which here comes
into play, becaufe the Logaritkms oa each Side of it
are known, viz. the Logarithm of 2 and 4.
10. Therefore (according to Theorem IV,) tf— 2^
fc=4, 2ZL. = ts =* 3, abz=A^ confequently iss=9,
and fo iss + tf^=: S = i;J Wherefore | + jx~
+
Chap, via hy Infinite Seriei,' $9
tHcn4^ g;=»xAxl5=B,tx^=fxBxii
3= C, &c. AMb i/f X T^j = f, WHertfore IVf +
L-^Jt ~ J^ 3- Thercfi>» L -^ +A+B4.C+D,
Igc. ss L 3. See the Opera^on following.
. L-^ = -^ ss . .. . : 0.4515449934959
. X J = A is .....;. . 255467342296
xxjs xAjt:B=s... .. 294656680
Thu5i
4xs-axC=t:D=...:; 15"
s
The Sum is the Ix^. of . . ... 3=o;477i2i2547*9-o
' 11; But this Logtmthm may yet much fooner
land with Ids Trouble be found, by the Artifice ufed
in Art. 7. For the Ratio -2^ = fifil = J, and
fo a4-^=s=65573, and <^--^=ss=37 » ^'^ ^"^
2^ X 4t = ^. therefore Li^i, 4. L 4, =
5x3' z" 3* 5x3* ' 2 '
L3»i butL^ = Lz^' — L5. AndL^
f L2"— L5= ;;.;.. i 3.81647993061
Therefore j 2«/ _. ; . , . . 0.00049010708
The Sum is the Logarithm of 3» = 3.81697003770
And * thereof is the Log. of3 = o.477i2i2547ii
thus the J?f^ Step of Theorem 11. gives the Loga-
rithm true to xi Places^,
6o Ofcdfi^ruSifigLoGAitiTBMi
12. The nextExample Hull bediat which Dr.
Haffey has ^vea for finding die Ijogarithm of 23,
which is done by Theorem IV. Indiis OUe^ 0=22,
^24, ls=:23, i8S=:529y 41^528, and is^ak=:
.0S7=S. Andf + jJ+^^. &c=iy|.
»" "5 - 73 - P "!«'^ ^ ^' +
L . '■ ', = L -r f andbecauie 2x2x2x3 =
24, aml2xix=22, therefore aiib 3L,a4-L3
= L245 andL2+Lii = L22. And^^^ld^
= L ^<ab I therefore, (proceeding in the Opera*
tion according to Art 10.) we have
■ — ^ ^" =s . . I.36i3i696u669o6i2945oo9i7a6698os
"p ssAsf. 41087462810146814347315S86368
7^ 7; ^ AsaBasa lasj 85 a 1 5441818^94^0074
^ X i X B=C= 65832351 8437617$
^^S''^*^^' 4208829765
' SS "■=^— »930
fie Sum = L 23 =r . . i.36i7?7836oi7592g78867777, ,2,5, ,7
which Logarithm is trac to 32 Places of Kgures,
and thus you may proceed for aoy other.
13. In making Logarithms for Prim Numbers^
the Artifice, otgreateft Mvantage tonfifts in finding
ixxh^Ratte or Framon, whofc Terms are thtgreatefi
poflible and their Difference the leaft j andthc Num-
ber whofe Logarithm is fought, or fome Power there-
of, IS an Ati^tot Part or Stdmehiple of one of the
Terms of the &id .Rati* or FraaioH. Fot thh once
obtaia'd, the Logarithm is (boa acquirad by The*-
rem
Chap.Vltt. ty Infinite Series, 6i
rem II. with cafe. Thus the Fraaion |Ui «
§iii2i »: i- } whetei=i, «&d 8=4641 the Series
will convei^ very fwift for the Logsrithm of '*'*"_
L29— Li I added to theSerien, gives the Logarithm
of 211. But the Raii^ or FraSion \Wltl. =ai
53 97040
tox54X29X43 ^'^ niakc the Scriesconvcrge very much
iboner than before ; for here dsi, and 8=10774081.
For L8o-hL54+L29-+-L43^Li2i added to the
Series (or Theor. II.) gives the Logarithm of 211%
half which is the Logarithm of2it. Laftly, the
'wcaon— y^rfJJTo— 60x18x55x113x197 = 7j^*^^
i/^i, and J=396423888i, converges to thatDegree
that the firft Step of the Series quotes the Logarithm
of the Fradion to 29 Places^ to which add the Lo-
^ithms of the 5 Numbers in the Denominatar^ and
It ^ves the L(^arithm of 21 1^, then i of that is the
Logarithm of 21I9 as before.
14. The greateft Difficulty confifts in finding out
proper Numbers for jAtxlucing fuch Fradions as
aforefaid ; and the beft Method of this is by prudent
fryals. An Example of which is here fubjoin*d.
Suppofe I would procin^e a convenient Fradion for
the Logarithm of 223, I ioiake tryal thus i '
Therefore 223 X387 = 64001
Having thus obtained the Term 64001, 'tis cafy to
ob&rve the other may be 64000, wherefore the Frac-^
tioo is imi » !!Z^^ and finds the Log^ithta of
223
6i Of conjlhi&ihg LogaiIithms
(223 with good Difpatch. Or thus, to find a Frac-
tion of larger Terms \ fuppofe I afliime the Nume-
rator 159000=1000x53x3, then to find the othct
Term as near this as may be, I try thus ;
223^ 3= ^^9
^CX 1 = 223
^^3lx 7 =1 561
Therefore 223 x 713 =ii 58999, which, is withiA
tJnity as great as the other Term 159000, and con-
fequently the Fraftion WHH is that required, and
thus you proceed to raife the Terms of any other.
.15. Let the Terms of ah)r Fr^dfidii be reprefented
by ^^^Leaft, and ^=Grcateft, Then if the Ratio ht
intreajing itv&llbo ^, but if decreafing, i.; let that
Term,- in which the Number fought is ingredient ^ be
cxprefs'd by the Produft rx, where r= the Number
(or Ptoduft of Numbers) whofe Logarithm is kriown^
and j^=: the Number whofe Logarithm is fbtight.
K a=cxj then ^ = 4> <>r "~ > \sat\ib=cxy thcli
ex
— • or —
J, or ^ ; alfo let there be put the, ad Theorem
?l+i^ + g + ii^ fSc. .= z ; then if the un-
known Number x be in the Denominatbr of the /»-
creafing Ratio jy viz. ^ j or in the Numerator of
the decreaftng Ratio ~, viz. — ; then it will be
Z+Ltf — L^=LAr. Andy vice verfa, ifitbe ^ or
A, the Theorerii will be Z+lJ-^hc=l^. Ftom
ifierice the Operations in the foregoing Articles for
making Logarithms haye their. Grounds and Reafon ;
4ftd
Chap. VIII. by Infinite Berks.' 63
and every thing there aflerted is from this Procefs
very evident.
16. To find the natural Number of any Lc^-
rithm propofed •, this is beft done by the Thcoreta
in Art 17. Chap. VII.
For Example, let it be required to find the Intereft o£
one Pound for one Day, at the rate of 6 /. per Cent,
per Jffnum, Coatpound Intereft % which is to extraft
the Root of 1.06 taken as the 365th Power; thus
Sic Logarithm of 1.06=0.025305865^2647702408-
46731 1 8635 1, ^c.
Whidi divide by 365, 1 £=0.00006933 1 1377 1 16-
the Quotient is J 592899910443^^6, (^c.
The next neareftl fc=|.oooi6=o.oooo6948i5587-
' Log.&itsN^l 28037517724712696, £«?<•.
Their Difference is 3=0.000000150421016338227-
733668350, fcff.
Mult, this by »=2.3025850929940456840i799i4-
54684, &c.
TheProduftis
»a =?= 0.000000346357189893416971322305 .
119963302990,864503
4i550»525H
14391
The Powers of n a. Then
i-f4«*a*=*i.ooooooooooooo5998i65i49543225i
/4»*a*;= 599
The Sum iH-i7»»S»4- \ t .00.000000000005998 1 6$-
A«.*a*=' 3 i49543285i==X. .
The odd Powers
{»a= 0.000006346357189893416971322305
i»5a"'= 6925025419
Sum »a-H»'3*=o.oooooo346357i 8989342389634.
7724=z.
Then
f »a =?=
64 7& Ijgarkbmk Sfdral.
Then the Value of the Series ia
X-«=o.99999965364287oo88227599o85 1 2^.
Which multiply bf ^==1.00016, produces
t»f
jnthsn L> and that to 30 Places of Figures. The
fime Number may be feen produced to 60 Places in
Mr. ^Amcra^Mathea^tical Tables.
CHAP. DC.
{^ the ^IjKtaritbmc ^Spiral I and the Nafun
0nd C§nfiruBion of the TaAk rf "^Mtridimal
*Parts^ or tbe Nautical ^Meridian *Line^
deduced therefrom.
i.T F any Right-Lim^ ^W be moved with an e-
X fioble Motion at)out the fix'd Point />, and at
the iame time the PcMnt W be movM towards the
Point /; with a Velocity fuch that the Radii pVf, pV^
fS» (^e. form'd thereby, be in a Geometrical Ratio
decreafing» then the Curve W VSQ» (^c. is called
the Logarithmic Spiral ; and diat for the £une Rea-
Ion as the Logaridimic Onrve before defcribM received
its Appellation. See Chap. III. Art 8, 9, la
a. For fuppofe the Arches A(S=£E=EG, dfc.
and therefore m Arithmetical Progreffion $ andfince^
from the Generation of the Spiral^ the Radius
pB:pD r.pD:pF::pF:pH»^c.*tisevidentthe Arches
AC» AE, AGt tSc. are the Exponents of the Ration's
oi^Raiii Dp» Fjpt, Hjp, 6fr. to thefirft pB ; and
fo thpf? Arches are m refpeft of the Radii ^ as Loga^
ritbwts VBLVtfyeSt of Numbers ; as is fufficiently mani*
feft from the preceding J}>eory of Logarithms. Where-
fore
Chapik. 7le LbgaHthfiic Spiral. 65
fbreifBpbc=i, 10, ioo,f5f^.aiidPW(thc lOthPro-
portiooalfroxn pB) be=io, IQO, 1000, &c. then (hall
the Arch AC=, 1000000, AE =,200000, AG =,
3000000, £sfr. AW=i ,0000000 ; be the Loga^
rithras of tte Numbers pD=i, 259, (^c, pF=i,585^
C^'r.pHzi:! ,996, fcf c. pWzrio j of Mr. Brigg^s Form.
3. This Spiral is alfo called the Equiangular Spi-^^
ral i becaufe it interieds all the Radif pW, pQ,pB»
at equal Angles. For fuppofe the Arches NP, TWi
infinitely fmall, and equal to each other, then may
the Parts of the Spiral DQ^and VW, be cfteemcd
Right-Lines; and i(b fince m the Triangles pOQ.
pV W, the Sdes are proportional, v/e. Op : pQ^:;
Vp : pW, and the Angle OpQ^VpW, thofe Tri-
angleis. are fimilar ; and coniequenily the Angle
pOQ^pV W^ br pQp=pWV 5 and thus it will be
every where.
4. Now let the lyhole Scheme Be cdnfidered a9
the Sterasgrapbic Projedicn of one Qjurtcr of a pe^
tallel Hemifpberey then fliall p be tbeP#/^ 1 WL A,
g Quadrantal Arch of the Equktor \ the RaMi p W,
pT, pR, iSc. the feveral Meridians projcfted on the
Plane of the Equator. And fince 'tis the Property of
tvery Rumb Line to make equal Angles with everf
Meridian oh the Globe, and the Angles contained
between circular Arches on the Globe, are equal to
the Angles between thjB fame Arches in this Projec-
tion, therefore the Logarithmic Spiral WQg is the
Projedion of a Rumb Line \ fince it has the fame
Property on the Projeftion, as the Rumb on the
Globe, as was proved Art. 3. hereof.
5. Moreover, fince all Right Circles^ fuch as arc
rtie A^ridiafts in this Cafe, are projeiSed into Rigbs
Lines equal to the Tangents of half the Arches, the
Lihes pB, pD, pF, pH, i^c. will here reprcfent the
Tangents of half the Complements of the Latitudes
AB, CD, EF, GH, ^c. And fince the Arches in
the Equator AC AR, AG, ^c. are the Differences
^ K • «/
66 7}>e Logarithmic Spiral
of Longitude made by failing from the Latitude B to
the Latitudes D, F, H, ijc\ on the RunA or Spiral
BOW; and it has been fhewn that thofe Arches
are the Logarithms^ o( the Radii ^D^ pF, pH, ^c.
therefore the Difference of Longitude is the i-^tf-.
Hthmof the Tangent of half the Complement of Lati-^
tudcy reckoning from the Meridian A p whence the
Logarithms begin.
6. Therefore the Difference of Longitude RT,
made by (ailing from the Latitude $ to the Latitude
y, is equal to the Difference of the Logarithms
(AT— AR) of the Tangents of the half Comple-
ments (Sp, Vp) of the Latitudes TV, RS. Andl
fince the Ratio of the Progrejfton^ or of pW to pV,*
may be infinitely varied, 'tis plain the infinite Num-
\xx of Rumbs in a Quadrant of the Compafs deter-
mine fo many Scales^ Logarithms in the Equator of
^he Tangents of the half Complements of the Lati-
tudes proper to thofe Rumbs.
7. Since then every different. Rumb is a Logarithm
inic Spiral J or determines a peculiar Scale of Log0^
rithms for the- Tangents of the Half-Complements of
its Latitudes,' therefore any Canon or Table of Lo-
garithm-Tangents, whether of Jfeper^s^ Briggs\ or
any other Form whatfoever, is the Scale of the Pif-
ferences of Longitude on fome determinate Rumb or
other. And confequently if this Rumb be invefti-
gated for the Canon of Brigg^s Logarithms (now la
common Ufe,) the faid Canon may be made to'an-
fwerall the Purpofes ot tht Nautical Meridian Line^
in Propofitions of Navigation by Mertatofs Chart. ^
8. In order to this it muff: be confidered, that the
Meridian Line is a Table or Scale of Longitudes to
every Degree of Latitude on the Rumb which makes
an Angle of 45 Degrees with the Meridian ; fince
in this Cafe the Differences of Longitude are always
equal to the Meridional^ or enlar^d Bifferemes of
Latitude. And fince there % a certain Rumb oa
which.
chap. IX. ^^e Logarithmic Spiral. 6j
which Neper^s or Brigg^s Logarithm-Tangents are
the Differences of Longitude, and the Differences of
Longitude on different Rumbs are to one another as
the Tangents of the Angles of thofe Rumbs with the
Meridian ; therefore by having given theDiiFcrcnce
of Longitude on theRumb of 45°, in Logarithms
of Nepet^s Form, and the Length of the Arch of one
Minute or Degree in Parts of the Radius, we can
dience find the Angle of that Rumb which deter*
mines that Species cf Logarithms.
9. Now the Momentary Augment or Fluxion of the
Tangent- Line of 45®, is exaftly doubJe to the Flux^
ion of the Arch oif the Circle (as is eafily proved),
and the Tangent of 45'' being equal to Radius^ the
Fluxion alfo of tht Logarithm-Tangent will be double
to that of the Arch, if the Logarithm be ofNeper^s
Form i but for Bri^s*s Form, it will be as the fame
double Arch multiplied into ^0.43429, &c. or di«
videdby »=:2. 30258, &c. the Index for Briggs*sLor
garitbms. See Chap. VL Art. 7.
10. l^Tow fince the Radius of a Circle being put
=*=i, the Periphery thereof will be 6.2831853, Gfr.
therefore 360)6.2831853, £9^^.(0.01745329, (^c. =s
the Length of the Arch of one Degree. Alfo
60)0.01745329, &c. (0.0002908882, &c. = the
Length of an Arch of one Minute^ in Parts of the
Radius. If one Minute be fuppofed Unity, then the
Proportion for finding the Angle of the Rumb
required for Neper^s Logarithms, will be, as
I : 2.908882, &c. :: Radius = loooooo, &c. : the
Tangent =2^08882, &c. of the Angle 71' 1' 4^'^
whole Logarithm is 1Q.463726117, &c. and under
that Angle is the Meridian interfe(^ed by that Rumh
Line, on which the Dirfercnces of Neper*s Loga-
rithm-Tangents of the Complements of the Latitudes
are the true Differences of Longitude, eftimated in
Minutes and Parts, taldng the &rft 4 Figures for
Integers.
K 2 lit
68 CofifirueimoftbeT^aik
%i. But fince Neper^s Logapthms are to thoA: of
Mr. Briggs\ Form, as 2.30.2585, &c. is to i .ock^ooQij
&c. therefore to find the Angle of the Rumb ifor the
logarithms of Briggsfs Form ; this muft be the A-
nalogy. As 2302585, &c. : 2908882, &c. :: loooooq
= Radius : 12633 114, &c* =^ ^^® Tangent of the
^"gl^fi" 38'9', whofeLogarithpx 1810.101510428,
&c. Wherefore in the Rumb Line that makes an
Angle of 51** 38' 9'' with the Meridian^ the commotit
(viz. -Br/gf j's)X#ogarithm-Tapgeiit§ are the true Pif-
fcrences of Longitude.
, 1 2. But if a Table or Scale ofJJgarilbm-'itangenis be
made by Extradion of the Root of the infiniteft Pirw-
eVj whofe Index i$ the Length of the Arch you put;
for Unity in the faid Scak •, then fuch a Scale oT I^-
garithm-Tangents fhall be the true Meridian line re-
quired. If then the Radius or Tangent of 45*, be
put = I ; and the Diflirence between Radius and
any other Tangent T, be called t; fo that it* be
R±t=T ; the Logarithm of the Ratio of Radius to
fuch Tangent, will be
Logarithm of the Tangent T, when it is R+t=T.
OrJxt+CH^ti +!! 4.1:1, (^c. when if:
2 3*^4^^5
IS R — t=T. All which is evident from Chap. VL
Art. 6.
13. According to the fame Dodlrine (Art. 9. of
the fame Chap.) if T be any given Tangent, and t
the Difference thereof fi^n^notj^LTangent ; thei»
the Logarithm of their Ratiowlnsf^ ^ r^-f*
\^ -ir- + jh^ ^'- ^^^'^ ^ « thel&eT
Term. But
when T is the greater Term, 14.
jphap. IX. ^Meridional Parts. €^
14. Again, it was (hewn in the ikmc CSiap. VL
Art. 10, and 1 1 . that thisScrics may be made to con-
verge twice as fwift, omitting all the even Powers^
by putting 5^= the 5'«w of the Tangents, and t as
the Difference J as above. For thus the Logarithni
garithm of the Ratio of thofe two Tangents.
15. But the Ratio of T to t, or of the Sum of tw0
Tangents to their Difference is the feme as that of the
Sine rftheSum of thofe Arches to the Sine of their
ipifFerence ; that is, again, as the Rado of the Ctf-
Sine of middle Latitude (or half Sum of the Arches)
to the Sine of half the Difference. Therefore putting
S = Sine-Complement oi middle Latitude •, and s for
the Sine of half the Difference of Latitudes > thea
i- = — i and fo the Series will become -; % L jl.
iL. X iL 4- iL , fc?r. wherein as the Differences
pf Latitude are fmaller, fewer Steps will fuffice.
1 6. So that, if the B/puitor be put for Middle-La^
titudcy then Ihall SrsRadius, and /=5ine of the La-
titude ; then the Meridional Parts reckoned from the
Equator will be ^ x f + il + 41 +^,,6?r.
Here bccaufe r=i, therefore ^x^-f-ll^-ii^
C, fffr. the half of which is i + '1 +.C+ C.^
0?r. = half the Logarithm of the Ratio of r +• j to
r — S'y that-i^yrfitb^ verfed Sines of the Diftances
from both Pole$. See Chap. VI. Art. 1 1.
i -17. I (hall exemplify this Series by ihewinghow
the Meridional Parts anfwering to 30* Latitude, wre
to be found thereby, and that by the LogU'ithms^
as follows.
The
Multiply by ..•.,.-.,.., *. 3
The Logarithm of , . . . j':?: .9.0969106
Subftrad the Logarithm of 3= 047712 13
There remain* thpi s^ ^^.,ic/:^^ q ^,/^l*Qo•.
Logarithm of I 7 = 0.0416657=8.6197887
Andprocecdingthus, [7 =o.oo62500=.7.79588oQ
you'll find the o- x7 ^ ^
ther Steps of the] ^=0.0011160=7.0476920
i. =0.0002 1 7I=:.6. 3364875
Series by their Lo-
garithms, as here
fct down.
^'— =o..oooo444=-5-6472773
TheSumofs-+- fl + T
f + r + gT + [=o-5492942=.97398o5V
ii:, &c. is \
11 J
To which add) * ^ « ,
the Log. of|i^34377467707g>&c. =3.5362739
TThe Sum is the-j
Log.oftheJk&- I
ridional Farts \ itfa:. 1888,334, &rc. =3.2760790
for the Arch I
of 30^ J
18. And thus you may proceed to find the Length
of any other Arch, or the Diftance from the Meri-
dian of its Parallel of Latitude ; ^nd fo the Meri-
dian Line may be conftruded de no^ifo^ if any one
thinks \t worth while. But tho* it may be dpne with
greater Accuracy and Exaftnefs by thcfe infinite Se^
ries than what wc have by the common Method ; yet
the Table of Meridional Fart 5^ or Nautical Lin^ ',
made from thence, now in Ufe, is abundantly fuf-
iicient for all the Purpofes of Sailing *, and confe-
quently
Chap. IX. cfMeridmalParts^ "^ f%
Guently renders a new Calculation thereof unaecef-
llaiy, and a Matter of mere Curiofity. And indeed^
fincc it has been (hewn above (Art. 9, 10, 11.) the
Meridian Line is no other than a Scale of the Loga*
rithm-Tangents of the Half-Complements of the
Latitude on the Rumb of 51*38' 9', the Propofi-
tions of Sailing by this Method are refolvable by
only the Canon or Log^rithm-.Tangcnts at the End.
of this Treatife ; fo that where this Canon is at hand,
neither Meridional Table or Line can be neceflary,
as will appear by a Chapter particularly on this SuIh
jeft, in the Prallical Part. They who would fee
the Thepry of this Branch of the Art^ may perufe
N**. 219. of the Pbilofopbical Tranfaffionsl where
they will find a moft learned Traft on this Subjeft,
wrote by Dr. HaMey 5 from whence the Subftance of
this Chaptier is taken.
C H A P. X.'
Oftbe Conftrufltlon of a Large ♦Logarithmic
Scale, €xbibiting by Infpe<aion a diftincS * I-
dea of the Nature and Agreement of^ Num-^
bers and their * Logarithms.
1. T T is an Obfervation of the earlieft Antiquity^
J^ that wc have no Ideas in the Mind whicK ^^ M
were not firft in the Senfes -, or that the Senfes of the, ^ ^
Body are the only Inlets or Entries by which the Idea*s^ ^. ^
qiObje£ls prefent themfelves to the Mind. It fol- •
lows then, that the Idea*s muft needs be fo much the
more clear and diftinSim the Mind^ and confcquently
be the better underftood by it, by how much the
mote fully y compleatly^ viud oivioujly they firft of all
affe(St our Senfes. Single uniform Objc(5l8 cafily in-,
^ linuate
ji TBe Conftrudlton ofd Scale:
llnuate themfelves, and make ftrong and clear Ini-
preflions on the Mind^ while thofe which involve
Multiplicity and Variety in their Nattire, are pro*
portionally more difficultly apprehended by thd
Senfes, and confequently afKsd the Mind with im*
ftrfeEly confufed^ znAJUghi ImpreJJims^ which there-
fore muft produce a moreperplex^d^ eh/cure^ and un^
eertain Noticn oi Conception of the Things them^^
felves.
2, From this G>nfideration we may eafily learn
tiieReafon^hy, of all the vaft number ofPerfonsi
who underftand the praSical Ufe of Logarithms, fa
very few of them know any thing of the Nature and
ConftruSfiott of them.. The Ufe of Logarithms is
rerv obviogs to the Senfes by eafy Examples, but
their Nature and Conftruftion lead the Mind too
much upon the Contemplation of Infinities both o^
^antity and Variety j which are Subjeifts too vaftly
^firufe znd remote from Senfej ever to be verv com-
monly underftood 5 unlefe fome Expedients oe con*
trived, which may help to facilitate fo difficult ail
Aflfair.
3, And as there are principally but three Ways^
whereby the Nature ot Logarithms are explained^
^iz. by Numbers^ Species^ and Lines^ the Expedient
albre£ud muft be fought in one of thefe three Me-
ibods. But Numbers^ of all things elfc, exhibit thfll
mod: complex and various Idea^ therefore it cannot
be hoped for from them. Species^ on the other hand^
«re too Jimple and concife a Reprefentation of fo vafi
and various Ideals as are thofe of Logarithms^ and
have nothing of their Refemblance in their Form i
tills Expedient therefore is not to be expefted from
diii Head. It remains then, that Right or Curvi
tints be ufed for the Purpofe of explaining the Na-
ture of Logarithms, by making the whole Matter ob^
viousto the Senfes. And here indeed we ihall ^xyi
ail that can be deiired, or is necefiary to the Purpofe.
Chap. X. ^e Con/iruaion of a Scale. 73
4. For Example, let the complex Idea of a tbou-
fond Units be to be exprefs'd moft advantamoufly to
a Mind unexercifed about fuch compleifd Notions ;
if you do it by Numbers or Figures, it muft be by
this Expreilion 1000 ; but there are but four Cba^
raSers to form an Idea of a tboufand Separate Objeds
in the Mind. In Species, this great and complefc U
dea is often reprefented by one Charader alone, as ^ ;
or two, as i-f-x ; which are ftill more obfcure and
abfblutely unintelligible without fome PrerNotions of
the Matter. But a Line may be taken of a Length •
fufficient, that by proper Divifions, all the thoufand
Units may be rendered diftinft and <^vious to the
Senfe, in any variety of Magnitude almoft, butefpe*
cially if they are equal to each other, as in the Cafe
of Logarithms before us. Wherefore, fince by Ldnes
fuch great and almoft inconceivably complex Ideas are
capable of being reprefented to the Senfes, diftina
and feparate in their proper Parts j and the DoSlrine
of Logarithms depending entirely on fuch IdeaSy 'tis
evident that by Means of Lines of afuf&cientLength,
the Nature and Properties of Logarithms, and the
Operations thereby, may be rendered more apparent
and compleat to the Senfes, and fo be better Mf^irr-
fiood in the Mindy than by any other Means what*
foever.
5. The Confequence of all which, is, that the
young T^yrOy and all who would have a true Notion
and moft clear Underftanding of this abftrufe and my^
fierious Doftrine, (hould be affifted with fuch a large
lineal ConfiruSion of the Logarithms^ as hath been
. hinted at. And this, I hope, I have efFcdlcd in the
large Diagram on the Copper- Plate before you, with
confidcrable Exadnefs, which I call the Logarithm
mic Scale. Wherein all that has been faid in the Ge-
neral Theory aforegoing, or may follow in thtpr api-
cal Operations of Logarithms^ is evident even to
Senfe itfelf^ to a very wonderful degree, by a bare
L4 Itilpedlion,
74 ^^ QmftruSHon of a ^cak.
Infpeftion, or a Glance of the £y^ only ; and th«re<«
fore cannot but conduce to form a very diftinA and
agreeable Idea or Notibn both of the Theory and
Praxis of this admirable Art.
6. The Scale confifts of three principal Lf^eji
which bound it ; thtjirjl is AB on the Side^ which
is 22 Inches in Length, and is divided into looo e-^
qual Parts which repreiient the naiurd Numbers from
1 to I POO, all which are viiible and diftinA to the
naked Eye % which Numbers therefore are affixed to
every ibth Diviiio^. The fecond is AC at the
3ottom, divided into 300 equal B^ts^ (as being but
I Bts Inches long.) Thefe reprefent the L^ariwms %
if each of thtftequ^I Parts be fuppofed to reprefent
10, or 100, the Logiirithmsy then^ for all Numbers
under 1000, will be exhibited by Lines only to 4
or 5 Places of Figures, including the Indicts. The
li&/y^ principal Line is the Logaritbm^tic Curve
cDEFB, in which all the Lines of iV«w*frjand X^-
^arithms terminate, and whofe Genefis and Proper.^
ties have been before 4cfcribed. See Chap. Ill, IV,
V, 6fr.
7. The Scale confifts (or is made up) of Lines of
Numbers, and others which are the Complements of
the Logarithms to 3.0QOO, The firft ^reperpendi^
cular to the Logarithmetic Line AC in its fcvcral
Divifions, and increafe in Length in a Geometrical
Ratio ; as hath been obfervcd : thus dividing AG
into 3 equal Parts CG=pGH,=HA, if GD be the
loth proportional Term from Cec5=i, or Unity, then
ftall GD=ioCe=io, and HE=ioGD=iooCe?s«
100. Laftly, ABf^ioHEs=iooGD=ioooCc=:
J ODD ; as is evident from the Nature of the Curve^
and by InfpeSiion. The Complements of the Loga-^
rithms are the Lines which run acrofs the Diagram^
parallel to the Line of Logarithms AC ; thcfe at the
Curve refer the Numbers to their proper Logarithms,
and by means of thofe Lin^a thus croffing each other
in
Chap.X* TieCcnJiruSionofaScak. - 75
iA every Part of the Scheme^ the Logarithms of Num-^
her^ and the NtLWibers oi l»aritbms art moft eafily
&hd obvioufly foundi for mt Extent of the Scale^
by Infpeaion only.
8. It is not pretended that this (or any other) In^
ftrument is capable of ahy great £xa)ftnefi in prciHtal
Operations \ 'tis fucflxient formy Deiign, ifitoiily
illuftrates void proves the Truth of every part of the
DoSifine of Logarithms to the Senfe^ and thereby
tenders it eaiier to the InteUeSs of young Learners.
If the Tbitnry before delivered be examined by this
Scale ^ it will be found to agree with it to zfenfible
ExaSnefs \ it being as it were but the fame thing at
large. In the following Parti I ihali (hew the Cor-
tefpdndente and matud Agreement bdtween the fun-
damental Operations by Logarithms wrote by Num-
bers^ and the fiune performed on this Scale ; than
Which nothing more^ that I know of« €an be faid of
expeAed.,
G H A p. Xt
Of the Conftru6tion f>f the Artificial hi^t^ bf
g/^/jJ^ Logarithms;
<
I. np H E Canon of togarittms being cotopleated
X «WJd orderly digefted in Books ^ tho* this was
a greater Adv.antage than the Mathematicians of any
former Age enjoy'd, yet not content to have a bulky
Book of Logarithms, fit tp be ufcd, in Studies and
with the Pen only^ the reftleis and unfatisfy'd Fa-
culty of Invention in Men put them upon Contri-
vances to nc^-model and reduce tljf voluminous Art
La to
^6 Cot^ruBion of Artificial Lines
to Mniature^ that fo it might be renderM more
eafily manageabk, and more univerfaUy ufeful.
2« la the Purfuit of this De(ign they alfo very
well fucceeded ; for (ihce Numbers of any kind are
capable of being reprefented by Right- Linesj they
were not long unappriz'd that the whole Body of the
Canon oi Logarithms might belaid down and ex-
prefs'd in tbie Diyiiions of one ftralt Line. Mr.
Gunter^ Profeflbr of Geometry at Gre/bam-College^
was the iirft who took this matter in hauid, and con-^
ftru^ted fuch an ^/f^/WL/»^ of Logarithms ; which
therefore from him was called (ever fince) Gunter's
Line^ orfimply, t\xtGunter. The (amePerfon alfo
conftrudted artificial Lines of Sines and ^Tangents ;
and ail thoie Lines, with fome others laid down on
a Scale, make what we commonly call Gunlet^^
Scale.
3. The ConJiruSion of thofe artificial Lines is
eafy to be underftood, and is as follows. Draw the
Rtght'Line AB (Fig. VII.) which divide into 10
great equal Parts, as is there denoted by i, 2, 3, 4,
&r. and •each ofthefe into 10 others, and foon.
Conceive thefe feveral Divifions^ or equal Parts^ to
reprcfent the Logarithms in the Canon for the natu-
ral Numbers. Now fuppofe the whole Length
AB=io, then the firft grand Diviiions will be i, 2,
3, 4, &ff. But if ABr=ioo, then the firft Divijions
will be 10, 20, 30, Csff. and the fecond Divijions
i> 2, 3, (^c. Agairt if AB=iooo, tht prime Di-
vijions will be 100, 200, 300, &?r. and thtjicondary
Divifions, 10, 20, 30, 6fr. Suppofe the latter Cafe,
viz. AB=iooo; then draw another Right- Line CD
e^al and parallel to AB, the natural Line of Lo-
garithms.
4. Now in the Line CD, fuch Divifions are to be
made as may reprefent the Places of the natural
Numbers i, 2, 3, &?r. or 10, 20, 30, 6?^*. or 100,
200, 300, &?f . But neglefting thtindices of Loga-
rithms,
Chap. XI. rfNumbtri^ ISines^ ficc. 77
f ithms, *us plain, fince the Logarithms of the Num-
bers i» 10, 100 % a, 20, 200 i 3, 30, 300 1 &c.
are the &me, the Dtftances of thofe Numbers will be
the fiime on the Line or Scale CD. And there&re
fince the Lc^rithm ofi, issaso, the Number i
muft be placed at the very Beginning of the Line
CD, from wheiKe the Logarithms begin in the Line
AB. Thenbecaufe the whole Line AB <ss 1000 tzr
Logarithm of io=CDy therefore againft the Loga-
rithm of 2, which is ^^sgoi in the Line AB, mSco
a Divifion in the Line CD, and by it place the Num-
ber 2. Again, becaufe the Logarithm of 3 is 2=47 7
in AB, therefore torrefpondent to d)e Point in AE^
m^c another DiviGon in CD, and by it place the
Number 3. The Logarithm of 4 is 602, therefore
from 602 in AB make a Divifion in CD, by which
you muft place the Number 4 ; and thus you pro-
ceed to find the Divifions for the other Numbm to
lO in the Line CD, by the Logarithms of thoie
Numbers in the Line AB.
5. If the Divifions in the Line CD now found
for the Numbers i, 2, 3, 4, &r. be fupposM, in«>
ftead of them^ to be for the Numbers 10, 20, 30^
40, &r. then each of thofe Divifions may be fub-
divided into 10 others, by the Logaritbmic Parts 10
the Line AB. Thus, because the Logarithms of
II, 12, 13, I4i 6?r. are 41, 79, 113, 146, iSe.
therefore againft the(e latter Numbers in the Line
AB, make Divifions in the Line CD, fo ftiall the
firfi: grand Divifion from i to 2 be divided again into
10 others. Agam, becaufe the Logarithms of 21,
22, 23, 24, ^c. are 322, 342, 361, 380, i^c.
therefore Divifions made in CD againft thefe Num«*
hers in AB will finifti the Subdivlfions of the Space
from 2 to 3, in the feid Line CD. And thus pre-
ceeding, you may fubdi vide all ih^ other grand or
prime DivifionSf to the £nd of the Line.
7S Cm^Bim i/ArtiJkialLinei
6\ If your Lines be of fd great Lengthy tluC Oititt
kft Sabdivifions in CD, aTfe ftill of aUngth capabte
of another tmfdd Divifion \ then the firft grand Di^
vifioos muft be reputed loo, 206, 366, &r. and fo
fince the Logarithms of the Numbers loi, 102, (Se.
201, 202, ^^. 301, 302, 6fr. arc 4^ 8, Cs?r. 303,
305, &r. 478, 480, &r. therefore ' if againft thefe
Parts in AB, you make Divifions in CD» there will
enfue a triple Divifion of the faid Line CD, which
. is more than is neceflary for Inftrumental Ufes^ ahd
indeed cannot be done but only for the two or three
firft Diviiions.
7. Thus have you (een the Conftnidtion of the
Artificial^ Logariihnic^ or Guntet^s Line^ fo famous
in all Parts of the Mathematics. A Line which per-
forms the Buiineis of the whole Logarithmic Canon \
iince the Divifions of this Line l»ve all the fame
Properties with regard to the natural Numbers on it,
as the Logarithms of the Table have to the Numbers
corrcfponding to them. *Tis plam the Divifions and
Relation of thefe two Lanes AB, CD, are the fame
as At, and AT, in Fig. IV. and III. Confequently
what has been ilaid of thofe Lines heretofore will help
to lUuftrate the ^eory and Conftruftion of the Lines
now under Confideration. But fince in Ufe the
Gunter CD isfuppofed to be divided into an lob
Parts at leaft, therefore you always (or moftly) ob*
fcrve it 6f a double Length of that which is exprcfled
in Fig. VII. which Length is commonly called 22^-
dius % and fo the Gunter in common Ufe is iaid to
be of a double Radius ; becaufe elfe the Divifions for
the nine Digits would be wanting, fince the Diftance
from I to 10 is eaual to that from 10 to 100, as is
evident from the foregoing Conftrudion, and from
the T'A^ory of Logarithms.
8. Having thus fliewn the Conftrtt3ion of the Line
cf Numbers y the Con/iruaicm of the Lines of artificial
iines and Tangents cafily follows j fince, as before
.. obfervcd,
Chap. XI. ofNumbiTSy Sines^ ficc. 79
pUerved^.the Logarithms of Sines ztidTangents are
nothing more thuk common Logariibmsoitacli Num«
|l)ers as exprefs the Sines and Tangents of each Minnie
gfitic^dranL
Ocg.
N.Sine.
1-g.
N.Tang.
L<^.
I
17
2418
17
2419
, 2
34
5428
35
5430
3
52
7188
52
7194
4
69
8345
1°
8446
5
»7
9402
»7
9419
6
|04
1.0192
105
I.02l6
7
121
1.0858
122
I.089X
8
>39
».i435
140
I.X478
9
156
i.«943
»58
1. 1997
10
^73
1.2396
176
1.2463
20
342
1.534®
363
1.5610
30
500
1.6989
577
1. 76 14
40
642
1.8080
839
1.9238
50
766
1.8842
1191
2.0761
60
866
1-9375
1732
2.2385
70
939
1.9729
2747
2^389
80
90
.984.
1000
1.9933
2.0000
Innn.
^'753^
Infin.
In the Hftle Table above, the firft Q)lumn contains
the Degrees J the 2d and 4th the Natural Sines and
Tangents^ and the 3d and 5th Columns contain the
Logarithms of tjiofe natural 5i»w and Tangents^ the
Indices being omitted^ and the Radius fuppofed ==
10000.
9. Let three Lines be drawn, and let L ^ Line of
Logarithms^ or double Radius of 20000 equal Parts ;
S = a Line for Sines j and T = Line for Tangents j
the two latter muft be drawn equal and parallel to the
firft i as in theConftruftion of the Line of Numbers.
Then having graduated the Line L mto 20000 equal
Parts,
8o ConJlfuBim of Artifitial Lines
Parts, ifagainft Aich (^thofe Parts as are exprefsM
by the Numbers in the 3d Column, you make Di*
vi(|9ns in the Line S, and by thofe Divifions you
place the Numbers in the firft Column, you will thefft
have the artificial Line of Sines S graduated for the
firft great Diviiions of i, 2, 3, 4, 5, 6, (^c. 10, 20,
30, Gfr. Degrees : After the fame manner by the
Table of Logarithmic Sines you find Numbers, fi-om
whence in the Line L you findDinfions in the Line
S for Minutes^ and Parts of Minutes. And thus the
Line of artificial or Logarithmic Sines is finiibed.
10. Again, from the fame Parts of the graduated
Line L, as are found in the fifth Column oif the Ta-
blety you make Divifions in the Line T, and by them
place the Numbers of the firft Column, the Line T
ihall be the artificial Line of tangents graduated for
the firft great Divifions of i, 2, 3, 4, 5, 6?f. 10, 20,
30, £5?r. Degrees. And the Subdivifions for Minutes
will be found as before direded. But tho* the doUile
Radius on the Line L fufiiccs for graduating the Line
of Sines S, to th6 whole Length of 90 Degrees, be-
caufe all Sines arc lefs than the Radius of a Circle,
which is the greateft Sine ; yet becaufe the Radius of
a Circle and the tangent of 45 Degrees are equal ;
therefore 'tis evident the Logarithms of all Tangents
greater than 45 Degrees, will exceed the Length of
the Line L, as is plain from the fifth Column of the
foregoing Tablet.
1 1. But fince Radius iszmean ProportionalhtVNt^ti
the Tangent of any Arcb^ and the Tangent of that
j^frriV Complement, itfolfows, that the ;?^/«r4/7W»-
gents in the Gecmetric Ratio or Scale zrt equally diftant
on ^ach Side from the Radius or Tangent of 45 De-
grees : and therefore the Logarithms of thofe natural
Tangents^ which are equidijlant en each Side the Ra-
dius or Logarithm of 45^, are alfo equidifferent ; that
is, their Differences are equal. Jhus the Difference
of 44^" and 46^, from Radius is the fame j and
the
Chap. XI. of Numbers^ Sines^ &c. 8r
the DiiFcrenccs of 40^ and 50®, 30^ and 60^, 20®
Mid 70^, fc?r. arc refpcftivcly equal to each other ;
and confequently the firft great Diviiions from i to
45^ on the Line Tofarftficial Tangents y will like*
wife ferve for the C^-Tavgenis of thofe Degrees, that
is, for all the Tangents from 45^ to 90^, reckon'4
back again to the beginning of the Line T« And
this is the Rcafon why on thofe Lines of Tangents^
you fee the Numbers placed at each loth Divifion,
thus iO|8«, 20I70, 30J60, 40|f09 45, attheEnd.
For otherwife the faid Line of Tangents, muft be
continued out to double tke Length it now is, which
would not be near fo convenient.
12. The Numbers in the 2d and 4th Columns are
the Divifions on the Gunter^ which correfpond to the
jfimilar Divifions qn the Lines of Sines and Tangents ;
wherefore the former being already made, the tw$
latter may alfo eafily be conftruftcd by means of that.
And thefe things are all I judge neceflary to be faid
here concerning the Conftrudion of thofe excellent
Lines of artificial Numbers^ Sines j and Tangents \
and as to their Ufcs, that will be a Subject to be
treated of after the fraificalUfe of the Logarithms
themfelves is firft explained and inculcated j for then
, the Ufe of thefe Inftruments will be much better con-
ceived and underftood.
^
M CHAP,
Si
'ne Manner ofujmg the
CHAP. XII.
(y/A^ Manner £^ufing /£v Tables g^Lo-
GARiTHMS in rRA.CTiCE \ andoj the Prc7
xtc^\SiXz% thereto.
i.fT^HE Logarithms being made for natural
X Numbers by Ibme of the Methods before-
going, the next thing neceflary was to difpofe them
into Tome convenient Order ox Form for praSicalUfes.
And fuch a Digeji or ColleiSlion of Logarithms, w^
call the Logarithmic Canon or ^Tables.
2. Thefc Tables arc of two Sorts ; the lirft coi\-
tains the Logarithms of all natural Numbers from U-
rity or i to loooo, or ipioop (as thofe large Tables
of Mr. ^herwin.) In thefe, the general Manner or
Form is fuch as here exprefs'd
in the Margin, which confifts
of two Columns ; in the firll are
placed the Numbers^ in the fe-
cond the Logarithms eorrefpond-
ing thereto, with their Indices.
And three tf thefe double Co-
lumns fill a Page in common
Books of this Form 5 and this,
of all others, is the moft obvious-
and cafy as to its Ufe, which therefore can need no
Explanation. For by Infpcftion only is feen what
Logarithm belongs to' any Number within the Com-
pafs of the Table.
3, But tho' the aforcfaid Foriff be the moft natu-
ral and obvious, yet it is not the moft artful and
comprehenfive ; therefore another Form or Difpofition
of the Tables for natural Numbers has been contrived
more
Numb.
Logarithms;
997
998
999
1000
lOOI
looa
1003
2.9986951
2.9991305
2-9995655
3.0000000.
3.0004341
3.0008677
30013009
V
chap, XIL Tailes of Log AKiTUMS. 83
more concife, or which takes up le(s room, and is yet
as perfeA and ufeful as the other ; a Specimen of this
I^'orm 1 have here anneited.
N<*. 1 Logarithms. ]
173
174
.'75
1 X
2
3
4
238046
240549
243038
238297
240799
243286
238548
241048
243534
238799
241297
243782
239049
241546
244030!
In which the natural Numbers are placed in the Side*
Column to the Irfi band^ all but the Units Place^ or
firft Figure of the Numbers, which is found in a pa-
rallel Column on the top of the Page, in the Order
*» i» 2, 3 4>^^. as you fee in the Specimen* By
this means one Column of Numbers fuffices for one
Page J whereas in the other Form there are three fuck
Columns, . the whole Page itfelf confiding entirely of
the Logarithms, which in this Cafe admits of five
Columns •, but the Indices m here omitted as not be-
ing ncceflary, fincc they are kiiowln by the Numbers,
The manner of ufing this Form is yet very eafy.
i^or Example, to find the Logarithm of the Num-
ber 1742 ^ againfl 174 in the Side-Column, and under
the Units Place 2 at the top, I find the Logarithm
241048, and fince the Number has four Places, the
Index mufl be 3; where 3.241048 is the Loga-
rithm compldit for the Number 1742. Thus the
Lo^rithm of , 17533^3.243782 ; and fo for others.
4. But the Tables of Logarithms are yet capable of
a further, ahd much more curious Improvement with
regard to their Cpntraftion or Concifenels 5 for fince
the Differences of Logarithms decreafe as the Num-
bers increafe, 'tis plain thofe will grow very fmail a^
thefe become very large j and confequcntly the two
or tbree firji Figures of the Logarithms to the left:
will be thtfame for divers large Numbers together in
M 2 the
84
^ Manner (fujing the
the Canon. Thus for Inftance, the Logarithms of all
the Numbers between 4168 and 4266, have the two
firft Figures to the left the lame in every one, viz.
62. So likewife all the Logarithms between the
Numbers 9954 and 9977 have their firft three Fi-
gures the feme, viz. 998, the Difference of the Lo-
farithms being only in the remaining Figures. The
'igures of the Logarithms therefore may be reckoned
of two forts, viz. fuch as are permanent or the iame,
for certain Intervals \ and iuch as are variable or al-
ways altering. In this Form of the Canon now\un-
der Confideration , thefe permanent Figures are
printed but once for their refpeftive Intervals, and
that in- the firft Column of Logarithms next the
Numbers the variable Figures only pertaining to each
Logarithm make the Subftance, or fill the wholc^
Face of each Page, as in the Specimen here fubjoin'd.
N^
Logarithms.
132
I
2 I 3
4
1888
12.0574
0963
1231I1560
133
3852
4178
45044830
5^5^
134
- 7105
7429
7752 8076
8399
135
13-0334
0^55
0977
1298
1619
136
3539
3858
4177
4496
4814
^37
6721
7037
7354
7670
7987
i;^{^
9879
♦0194
0508
0822
1136
^39
945
14-3015
3327
478
3639
524
3951
570
4263
616
975^3^
946
,891
937
983
*029
075
947
976.350
39^
442
487
533
948
808
854
900
946
991
949
^yy.266
312
■35^
403
449
. 5. The Numbers here, as in the laft Rfrm^ are,
for the three firft Figures to the left, found in the
Side-
Chap. XII. J'ahUs ^/Xogarichms. 85
Side- Column, the other Figure at top. ThcLogft'
fkhms in this Specimen are 65, yet but fix of thwn
need be cxpreffed at length, viz. thofe for the Num-
bers 1320, 1350, 1390, 9450, 9470, 9490. The
Logarithm for 1320 is thus wrote, 12.0574, to de-
note the two firft Figures 12 (feparated by a Dot)
are permanent thro* the Interval between 1320 and
J 350, that is, they belong to the Logarithms of all
the internaediate Numbers between thofe two, and
therefore need be exprefs*d only for the firft y the
other Part of the faid Logarithm which is variable,
is exprefs'd alone in all the reft. ., The upper Part of
this Specimen or Tariff conC^ of Logarithms having
two Figures in the permanent Part i the lower part
is an Example of Logarithms having three Figures
in the permanent Part. To find a logarithm there-
fore to any given Number, will alio be very eafy ia
this Form. Thus, fuppofe the Logarithm be fought
. for in the Number 1323, 'tis found in this Manner*
Take- the permanent Part either againji or next above
the three firft Figures 132, which here is 12, then
againft 132 and under 3 at the top, you find thex;^-
ri able Part 1560, to which prefix tht permanent Part
12, and you have the Logarithm 121 560, which
with the Index J is 3. 12 1560. Again, to find the
Logarithm of 1374, take 1 3 the permanent Part next
above the three firft Figures 137, then againft 137^
and under 4 at top, you find the variable Part 7987,
which annexed to the former Part 13, make 137987,
and with the Index^ 3*^379^7% ^^^ Logarithm
fought.
6. Thus alfo you proceed when the permanent
Part confifteth ot three Figures^ where the Intervals
are much ftiorter. One caution only is neccflary,
and that is, that you obfervc ih thofe Lines where
there is found an Jfterifm^^ to join all the pariabli-
Parts in that Line before the *, to the permanent
Part next above 5 and all after it, to that next ^elow.^
'* Thui
86 tbelianfikr ofujtk^ ite
Thu^ in Che 7th Line^ aiid id Column, )^ou feeiri
jifterifm ♦, therefore the variable Part ^Zj^ beforq
it muft be joinM to r^ tht permanent Part next above
it, to form the Logarithm 3.13918751 for the Num-
ber 1380: ^Mt tht variable Parts 61^^ 6568, &ff.
following it^ arc to be annexed to 14 the permanent
Part next below^ to form the Logarithms 3. 146 194,
3.140508, tfr. for the Numbers 1381, 1382, (^c.
The Reafon of which I prefume muft h^felf -evident
to every Reader.
7. I have Been ihe more prolix on this l(fi Forfk^
left any Obfcurity or Uncetiainty (hould remain to
difeourage or ptejudicfe tht young Tjro againft fb rari
and fo advantageous a Contrivance. I fay, fo rare ;
becauie I have never fcfett (amongft many) above ohe,
viz. Sberwitfs Canon, in this Form ; and that; by
reaibn of its great Price^ is not very common; Tho*
that Gentleman (ays in his Preface, he has ih this ex-
cellent Method followed Dr. John Newton in his 2r/- .
gonometriaBritannicaj a Book which I have notfeen.
The Advantage alfo of this Abbreviation is next to
that of the Invention it fclf ; for hereby the prolix
and unwieldy Tables (in their original Form) are re-
duced or abridged to one half the Bulk nearly ; all
thefuperfluous Part being omitted, and nothing but .
what was ncccflary retained in this Canon.
8. According to this moft excellent Abridgment
therefore, I have firft of all, thlt I know ibf, pub-
lifhed the common. Canon of Logarithms forNum*
bers from i to loooo ; having taken the no fmall-
Pains of tranfcribing the whole with my own hand
from the aforementioned large Work of Mr. Hen.
Sherwinj which is the moft correft of any extant.
9. In this Form or DifpofitJon of the Canon, I
have alfo publifhed the Logarithms of Siiies and Tan-
gents \ which thing hath not been done before in any
Work great or fmall," that I have ever feen or heard
©f. This makes the 2d Part of the Logarithmic
Tables,
Chap. XII. Tables ^/Logarithms. 87
Tables^ as mentioned Art. 2. And fince they are
here in adifFerent Form from all others, it may not
b^ unn^freflary to hint to the young Learner, that
^he Numbers exprefs'd in the Side-Column are the
Degrees, and every xoth Minute, and the Numbers
in the parallel Column at top are the Minutes between
the loths^ fee the following TariiF of the Logarithms
of Sines in this Form for the Minute^ from 72 De-
grees to 73.
"dT"
72. o
10
20
40
59
978.206
615
979.019
420
816
980.208
3 4
247288329I370
655\^96 7 3^.777
059 100 140 180
+60500539579
« 55^5 934-97 3
247l286J325,364
'!('o give an Example $ ^t the Logarithm be fought
for the Sine of 72^ 43^ Seek in the Side-Columi^
72^ 40', next* above which is the permanent Part of
the Lc^arithm 979 in the firft Column of Loga«-
rithms ; then a^inft 72^ 40, and under 3^ at top
you find the variable Part 934, whic^ annexed to
the other makes 979934 ; to which prefix the Index
(which is fet at the top of each Page) 9, and the Lo-
garithm is compleat, viz. 9.979534 for the Sine of
72° 43' » and thus you proceed for any other.
10. I have contracted the Logsfit^ms to iix Places
of Figure; 6n]y, as being fufficient in. common Ufe -»
th? natural Sines and Tangents ar? not h^re infertedi^
lince when their Logarithnis can be ufed, they tJicm-
^elves are ufelefs. Befides, whenever they are re-
quired, they may be imrnediatcly had from their
Logarithms. For Exan^ple, fuppofe X would know,
the natural Sine zt^d Tangent for ^S^ 47^, the Loga-
rithm, Sine and Tangent of this Arch, arc 9.796836,
and
S8 The Manner rf ufing the
and 9.965609; To fchcfe Logarithms (neglcfting their
Indices) find the natural Numbers, by the firft Part
pf the Canon, they will be 626377 ^^^ 803542, the
natural Sine and Tangent fought. (See the Method
below. Art. 15. for ftidmg the Number of a giveii
Logarithm.)
11. The Logarithm of the Secafis of any Arch, as
of j8®47', is thus eafily obtained :
From the double Radius 20.000000
Subftraft the Co-Sine of 38^ 47' 5= . , 9.891^27
There remains the Log.Secant of 38^47'= 10.108 1 73
And thus the Logarithm of any other Secant may
be fquod, and confequcntly the natural Secant^ }ox
natural Number belonging thereto.
12. The Reafon why the Indices^ on tjie top of
the Pages, of the Logarithms of Sines and Tangents^
are ib large, viz. 7, 8, 9, 10, 1 1, fc?^. is becaufe
the Radius of the Circle was fuppofcd to confift of
1 0000000000 equal Parts, whofe Logarithm there-
fore is lO.oooopo ; wherefore a Number of fuch e-
qual P<?r/Jexprcfliiigthe Sine of one Minute i', will
confift of 7 Pieces, whofe Logarithm then ^ill have
its Index 6. The other Sines will confift of 8; 9, and
10 Places, and fo the Indices of their Logarithms
will be 7, 8, 9, as in the Tables ; thus alfo the
Numbers expreffing'the Tangents, in fuch equal Parts
will confift of 7, 8, 9, 10, It, 12, 13, and 14
Places of Figures, whence the Indices of their Loga-
rithms muft be 6, 7, 8, 9, 10, II, 12, 13, accord-
ing to Art. nth and 12th of Chap. I. But firice,
as before faid, the firft fix Places of the Logarithms
to the left are fufBctent, the reft are rejefted as fu-
perfluous. ^
13. In future Operations there will be frequent
Occafion for what is called the Arithmetical Complex
ment of a Logarithm, which is nothing but the I)//-
ference '
Chap. XII. Tables of Logarithms^. 80
ference between that Logarithm and Logarithm-Ra-
dius id.oooood.
Thus if from 10.000006
You fubduftthe Logarithm 4.877026
There will remain the Arithmet. Comp. = 5.122974
And this is done mentally in an Inftant, by taking
every Figure from 9, except the fifft, which you take
frorii io;
Note. If the Logarithm be of any Sine or Ian-
gent, add 10 to tht, Index of .the Arithmetical Com-*
plement, and it will be the Logarithm of the Co^
Secant of the fame Arch. For Example,
Suppofe the Log; Sine of 3^ 48' = . . . 8.821342
The Arithmetical Comp. thereof is . , . i . 1 78658I
to which add 4 . . . ^ 4 . . i lO,
The Sum is the Log. Secant of 8^° 1 2' = 11. 178658
Which is evidently the fame Operation as that in
Art. II. hereof.
14. From the theory oi Logarithms we learn, tbaf
the Differences of great Numbers are proportional to
the Differences of their Logarithms. (See Chap. V.
Aft. 6, 7.) Therefore tho* the Canon of Logarithms
gdes no farther than the Number 1 0000, it may by
this means be extended to the Number 1 0000000,
or the Logarithm of any Number under 1 0000000
may be found by the prefent Canon^ according to
the following Rule.
Firft ; find the Logarithm of the four firjl Figures
6f the given Number, by the firft Part of the Canon.
Secondly ; fubftraft this Logarithm from the Lo-
garithm next greater or neS!t following in the Table j
and referve the Difference.
Thirdly ; multiply the Difference^ by the remain^
ing Pigiires of the given Number 5 and from the Pro--
du£l cut ofF to the Right handy fo many Figures as
there werfe remaining in the givtn Number.
N Fourthly' i
"1
9© 7he Manner ofufing the
Fourthly ; add the Remainder of the ProduSl to
the Logarithm firft found, the Sum ihall' be the Lo-*
garithm fought.
For Example, let the Logarithm of the Number
127053 be fought.
The Logarithm of the t ^^^ o
• firft 4 Figures } ' • "7^00 = 5.103804
The next greater Log. isof . . 127100 = 5.10414^
The Differences 100 342
Wherefore fay, as 100: 342 :: 53 : 181,26
53
1026
1710
181I26
Add the firft Logarithm 5.io;?8o4
The Sum is 5.103985 '= the Loga-
rithm fought for the Number 127053.
Example 2. Required the Logarithm of the Nuxn-
bet 3567894?
The Log. next following is of 3^68000 = 6.55242 5
The DifF. of Numb, and Log. 1 000 122
Th^n fay, as 1000 : 122 :: 894: 109,068
Add the firft Logarithm . .6.552303
theSumisthe Log. 3567894=6.552412, as was re-
quired. And thus you proceed for the Logarithm of
any other greater Number than thofe in the Canon.
15. By a Method reverfe to the foregoing, you
find the Number correfponding to a given Logarithm%
thus, fuppofe the given Logarithm be 3.567026,
and you would know the Number thereof. Seek
this Logarithm in the Table, and becaufe you there
find it exaSlly^ the Number 3690 correfponding
thereto, is the Number fought. But if the given
Logarithm
i
Ghap.XII, Tables g/' Logarithms. 9!
Logarithm be noUxaSily contained in the Table, and
more than 4 Figures be required, proceed as follows.
Firfk ; feek in the Table a Logarithm the next kfs
to the given one, for the four firft Figures of the
Number fought.
Secondly \ fubftraft this Logarithm from tht given
cne^ and annex to the Remainder, ^^ many Qypbers as
you feek Figures more than four.
Thirdly ; take the Difference between the Loga-
rithm juft found and the next greater^ by which di-
vide the faid augmented Remainder ^ the ^otient an-
nexed to the four firft Figures ftiall compleat the
Number fought.
Example i. Let there be fought theNumber to the
Logarithm 5.103985, to fix Places of Figures.
The Log, nextlefs is of . . . 127000 = 5.103804
The given Logarithm . . . , 5.103985
The Difference or Remainder 181
The next greater Logarithm 1 2 7 1 00 = 5.104146
DifF. between the greatefi and leafi Log. 342
Since the Places iu the Number fought are 6, aug-
ment the firft Remainder 181 with two Cyphers^ and
it will be 1 8 160 i then 342)i8ioo(=53, which an-
nexed to the four Figures 1270 before found, make
the Number 127053 required.
Thus alfo to the given Logarithm 6,55241 2, you may
find its proper Number 3567894, and foibr others.
16. In the fame manner you. proceed to find any
Pecimal Number to a Logarithm given, only in this
Cafe the Indices of the Logarithms are negleded rill
the Operation is finifhed, and then fo many Figures
are to be cut off from the Number found for Deci-
mals, as the Index of the given Logarithm Ihall indi-
cate ; what is here faid relates to plain or termi-- '
nate Decimals only ; but there are other forts of De-
<:imals which circulate or perpetually repeat one or
N 2 xpore
'94 ^e Manner ofujing the
more Figures ad infinitum : Andthofe Figures which
thus circulate zrt Repetendsj as in thefe Numbers,
viz. 235,2222, £5?r. 27,83333, ^c. 2.383838, &c.
702,6026026, fcff. 0,2672326723, fcfr. Now thefe
Repetends need be wrote but once if we flur the firfli
and laft Figures in each, to denote them fuch, as
thuS235,A 5 27,85- ; 2,?* ; 70^,60 ; 0,;2;6725', 6ff-
See more in my Univerfal Syfiem, or Body of Decimal
jiritbmetic^ printed for Mr. Noon.
ly. The Logarithms for the Re- [" ^=0.045757
peating 9 l^igits arc made by adding ^=0.346787
the Arithmetical Complement of the 5'=i:o.522879
Logarithm of 9, to the Logarithms 4=0.647817
of the faid Digits, and are fuch as< ^=0.744727
^=0.823909
7=0.890855
«=o.948847
L ^=1.000000
here annexed. The Lqgarithms of
pure compound Repetends are ipade by
adding the Arithmetical Complement
oifo many ^\ as there are Figures in
the Repetendi to the Logarithms of thofe Numbers
confidered as terminate. Thus the Logarithm of the
Repetend 56, f is found as follows :
To the Logarithm of 36.5=1.562293
Add the Arith. Complement of . . 999 =0.000434
The Sum is the Log. of the Repetend 3'6,f=i. 562727
and thus proceed for others.
;8. If the Repetend have any prefixed terminate
Part ; then from fuch a mixed Repetend fubftraft its
terminate Part^ and to the Logarithm of the Re-
mainder add the Arithmetical Complement of tbp
Logarithm of as many 9's as there are Figures in the
Repetend. For Example, fuppofe you would fina
the Logarithm of tht mix*d Kepetend 26S'92,7, pro-
ceed thus;
Chap.XIL Tables (>/ Logarithms. ^^
From the Repetend 26892,7
Subftradl the terminate Part 26
To the Log of the Rem. . . 26890,1=4.429592
add the Arithm. Complem. of 9999 =0.000043
The Sum is the Logarithm of 2 6«'92, 7=4.429635
JVJ7/1?, the Indices of the Arithmetical Complements
are here (as in thefe Cafes they always muft be) o-
mitted.
CHAP. XIIL
Of the Oxigmand Conftruaion g/'SHAKERLY*i
^«^Street'5 Logistical Logarithms.
^*. fTT^H E Ufe of Logiftical Logarithms is in AJlro^
X nomical Calculations^ or Sexagejimal Arith^
metic •, but this fort of Arithmetic, which taught the
Rules of Addition^ SubJiraSiionj Multiplication^ Di-
vifiony &c. of Semgefimal pra£lionSy viz. Degrees^
Minutes^ and Seconds of Motion or ^ime^ was, xafor^
mer timesj called Logiftical Arithmetic, And fince
the Invention of the cornf^on Canon of Logarithms^
Mr. Jeremiah Shakerly^ in his TabuU Britannica^
firft contrived from them a fort of Logarithms ad-
apted to the Rules of Logiftical Arithmetic ; and
therefore gave them the Name of Logiftical Loga^
rithms. And fince him Mr. Thomas Street^ in hb
Aftronomia Carolina^ has invented another and more
convenient Form oi Logiftical Logarithms ^ than Sha-
kerly^s. And fince Tables of both thefe forts of Lo^
giftical Logarithms are extant, 'tis proper to acquaiint
the Reader with the ConftruSion of both, which is as
follows.
2,
$4 Of Shakcfly'i and StrcctV
2. Since Logiftical Logarithms are altogether con#
cerned in working Proportions of Degrees^ Minutes^
and Seconds^ and more efpecially of Minutes and Se^
£onds^ t<^ether with Integers^ *tis evident, if thofe
Sexagefimal Fra6Hons were reduced into the loweji
J)enominationj viz. of Seconds ^boc. they might then
be work'd with the Logarithms of common Numbers.
Thus fuppofe the Proportion be 6d oo^ : 3' 47'' ::
51^ 29^ : 3' 15^5 if thefe/r^<^/^»^j/ Numbers be re-
duced to Seconds y they will ftand thus, 3600^ .-227^
:: 3089^ : 195'' ; wherefore *tis plain, the Propor-
tion in this Cafe mfiy be wrought by the common Ca^
non of Logarithms^ as will be hereafter fliewn. But
then as here is no Radius^ there will arife double
Trouble in the Work by Logarithms in common Ufe,
in firft adding the Logarithms of the two middle
Terms^ and then fubJiraSing the Logarithm of the
^rji from that Sum^ in order %to have the Logarithm
of the fourth Term fought ; or elfe the Complement
Arithmetical oi thtljyg2Lx\\}\m of X}citfirji Termmuik
be taken to perform all by Addition only. To avoid
therefore the Trouble attending perpetually either of
thefe Methods,
3. Mr. Shakerly makes this Proportion, as 3600^ :
227''^ :: 1 00000, ^c. : 0.06305, fc?r, or with the
Logarithms, thus 5
As the Logarithm of 3600'' s= 3.556302
to the Logarithm of 227''' = 2.356025
fo theLogarithmof Radius i.ooooo,&c.=io.oooooQ
to the I^ogarithm of o.o6305>&c.= .8.799723
Now 'tis plain, the two lafl Logarithms perform the
feme as the two firft ^ their Properties being the feme ;
but the firft of the two latter Logarithms is Radius
lO.ooooQO, which therefore he called the Logifticat
Logarithm of 3600''' or one Degree, and confequently
the Logarithm .8.799723 is the Logiftical Logarithm
of 2 2 yyz^' ^f. And thus the Logifiical Logarithm
of
Chap. XIII. i^i^/w/LoGARITHMS. iJ5
of any other Degrees^ Minutes and Seconds may be
found ; viz. by reducing them to Seconds^ and then
by taking from the Logarithm of thofe Seconds the
conftant Logarithm 3.556302 of one Degree in 5^-
€onds.
4. Thus from theLog. of. . . I'zzSo^-rzi.y^j^i^i
Take the Log. of one Degree =3600^^ =3.556302
therercmains theLogift. Log. of i'=6o''=*8.22i849
i^ 3' 00^=3780^= 3.57749^
Again, from the I
Log. of y
tzke the conftant Log. 1^ =3600^^ = 3.556302
X^riTof""} '" 3' oo'=378o'=.o.o.„so
By thefe Examples you eafily perceive how the Lo-
gtftical Logarithm may be found for any Number of
Degrees J Minutes and Seconds^ in Shakerlfs Form.
Andfince there are 60' or 3600^ in^^ Hour^ as well
as in one Degree^ therefore a Table of thefe Jjogiftical
Logarithms ierves equally as well in the Computation
of ^ime as Motion.
5. Butif theTimeofa whole
Day or 24 Hours be the Inte-
ger^ fince there are but 1440'
therein j and 3600^': 1440' ::
2'^i: i'; alfo fince 60' : 24HO
l:2'i : iH^ ; therefore if, thro*
the courfe of thefe Tables of
Logiftical Logarithms^ againft
every I'i you place the Hours ^
and againft every 2^'i (or rather
every 3d and 5th Second) you
place the Minutes of an Hour ;
the Table of Logiftical Loga-
rithms ioT Motion andTiW will
be compleated ; a Spedmen of
ivhich, in this Form of Mr. Sba-^
kerky%
° 'Motion
Time.
II 4^
H .
H
I
2
3
4
5
6
7
8
9
10
2700
987506
987522
9'^7SZ'^
9^7554-
987570
987586
987602
987618
987634
987651
gSy66y
XVIII.
. _ _-_
I
2
3
4
'p6 O/^Shakerly'j and Street*^
kerlef Sj I have before annexed. The firft Cdliiniti
contains the Degrees, Minutes or Seconds -^ the fe-
cond Column the Logiftical Logarithms thereof; atld
the third Column contains the Minutes of Time, the
Hour being exprefs'd at the top, viz. XVIII. aft-
fwering to the Motion of 45', or 2700'''.
6. But becaufe thefe Logiftical Logatitbms of Mr,
Sbakefley, confift of many Figures throughout the
Table, it minifter'd occafion to Mr. 7*homas Street to
contrive a more compendious and convenient Form
of thefe Logarithms ; and fuch he invented, which
tho* large at the beginning of the Tabk, yet imme-
diately leflen very faft, and fo continue to the End
of the Table, or 6d oi^6o& ; whofe Logiftical La^
garitbm is =0. / .
7. The Reafon of "vfrhich is manifeffi from tKfe
Manner of their Conftruilion, which is as follows';
Suppofe any Proportion of Sexagejimal Numbers^ as
that before made ufe of. Aft. 2. viz. 60' : 3^47'^ ::
51' 29^ : 3' 15^, whidh reduced to Seconds, ftands
thus, 3600^' : 227''':: 3689-^: 195^^ Now in ordet
to obtain a vacant ^erm in this Analogy, Mr, Street
(inftead of Sbakerly^s Analogy 3600'^ :22y^::i 00000 :
6fr.) inverts the Jirft Ratio, as thus ; 227^^^ : 3600'^
:: Unity ; to a fourth Number whofe Logarithm is
reputed the Logiftical Logarithm of the firft Term
227'^^, as the Logarithm of Unity is ofthefecond
3600^^^ See the Work.
The Logarithm of ... . 3' 47''^=227'''=2. 356026
The Logarithm of ; . i°=:6o'=z^6oo^=3.556302
The Logarithm of Unity 0.000000
The Logift* Log. of . . . 3' 47^''=2 2 7^^=^1200276
8. Whence it evidently appears, that to find the
Logiftical Logarithm of any Number of Seconds, yoU
need only fubftrad the common Logarithm of the
Number of Seconds from the conftant common Lo-
garithm of 3600*, for the Remainder or Difference
will
Chap. XIIL ?2iiJ5pi5f Logarithms. 97
Wifl be Ac ZtgiJtkalJL^ariibm requiitd, in S^e$^9
Fomi.
Thua from the Logarithm of, • 3600*^:2:3.556302
Subftrai^ the Log. of £t' 2^''=3o89 =3.489818
There remai» the Logift. Log, of 5 1' 29^= =^. ,66481
Agun from the &id Log. of. . 3600^=3. 5563 q^
Subdufl: the Logarithm of 3' 25^=195=2.290035
There remains the Logift.Log. of 3' ts^^slziSzij
9. From wheace *tis evident, that the greater the
Kumber of Seconds is, the lefs will be the I^x^f-
tjcal Logarithm thereof ( till you come to the Num-
ber 3600^ whofeLogiftical Logarithm is nothing at
«1], gs before faid. And thus it appears that the Ixk
g^ithmA of this kind of Mr. Sireei^s Form, have la
them fewer Places of Figures, and are therefore more
convenient forUfe, by much^ than thofc of Sbakerly^s
Form, before defcrib'd. And for that Reafon I have
4chofe to give the Reader a Table of Street*^ Loga-
jithmsmtherthan the other; and tho* Mv.Leadbetter
has gjyen us Tables of both forts, yet I think it in*,
tirely needle& ^ fince all the principal Uies of ^ibif^
hrfy% are piuch better performed in Streets LogiA
tical Logari^ms.
10. In fhewjng the Manner of making thefe Lo-
garithms from the common Ones, I have exprefled
tnematiarge,v/2. 1200276, 66484,1266267, the
Log^ftical Logarithms of 3^47^ 51' 29^, and 3' 15^,
as p^r Art 7, 8, Yet h^re two things are to he ob-
ierv*d: Firft, that the Index is not diftinguifticd
from Logaritfim itfelf, with a Pointy as in the com-
mon ibrt I but the remaining Figures, both of Lo-
g^thms and Indices^ be they more or lefs, are re-
puted toge^ther, the Lpgiftical Logarithm. Secondly, .
that two Places of Figures to the Right Hand in the
Examples, are ftruck offin theTabfe ; the other be-
ing fully, fufficient for all the Purpofes thereof. So
O that
9? O/'Shakerly'i and StrcctV
that m the Table you will find the &id Logifiical Lo-
garithms vfwtt 12003, 665^ 12663, &c.
11. Mr. Leadbetter has taken the pains to continue
the Table of Logiftical Logarithms in Street's Form,
to i2o\ or 2 Degrees ; but as there is little occafion
for any more than the Logiftical Logarithms of 60'
or I Degree ; and when there is, the fame Logarithms
are capable of anfwering it, 1 have therefore con-
tinued them no farther than the Inventor did, viz.
to 60' or 3600^
12. As to the Form of the Table, •tis very cafy
to be underftood, efpecially if thofe of the common
Logarithms of Sines and Tangents before defcribed
are : The firft Column of the Tables contains theD^-
grees or Minutes^ or Minutes and Seconds in Sexa^
gejimals^ as in the common Tables^ in the Order of
Denaries or lo's, the nine Digits running along on
the top of the Table, under which, in the; feveral
Columns, are the Logiftical Logarithms abbreviated
in the fame manner as thofe of Sines and Tangents ;
and are to be taken out according to the Direftions
there given, which fee. In the laft Column are con-
tained the Numbers of the firft reduced to Minutes
or Seconds ; and are to be compleated likewiie with
the Digits on the top of the Table. Thefe are re-
ferred to when the Logiftical Logarithm of any inte-
gral Number is fought. An Example or two will
render all eafy.
13. Let it be required to find tht Logiftical Loga^
rithm of 2^ 48' or 2' 48^ Firft feek 2 40 in the
firft Column, and agaidft it and under 8 at top, you
find 310, which annexed to its proper permanent Part
13 in the firft Column of Logarithms makes 133 10,
the Logiftical Logarithm of 2 48, as required. Thus
the Logiftical Logarithm of 37' 59^ is found to be
1986 5 and of 54' 40'' to be 404 5 and of 58' 37^ to
be loi 5 and of 59' sf to be 8. And thus the Z.^-
gijlical
Chap.Xm. Talks of Log AKiTKMS. gg
giftical Logarithm of 60', is =0 ; and fo will be a va-
cant Term in all Analogies for Operation.
14. Let it be requit^ to find the Logiftical Lo-
carithm of the Number 584. Firft feek 580 in the
kft or right-hand Column, and againft it and under
4 at top you fee *99, which ftiews it muft be join'd to
the following permanent Part y% in the firft Column
of Logarithms, and therewith makes 7899, the. Lo-
garithm fought. Thus the Logiftical Logarithm ^ of
1000, is found 5563 i and of 33599 to be 301 ; and
of 3596, to be5 -, and that of 3600 is nothing.
15. This Table alfo equally ferves for Time, whe-
ther for z Day z,nd Minutes^ or Minutes ^nd Seconds -^
by help of the little Table at the End of the Lo^
giftical Logarithms J which fhews what Parts of Mo-
tion in Degrees and Minutes correfpond to Time in
Hours znd Minutes. Thusagainft 13^ in Time, you
fee 33' of amotion j againft i Hour is 2^ 30' ; and the
Motion aiifweringto VIP 43', is 17^ 30', -f" ^^ 4^'
= 19^ 18' 5 and that anfwering XXP 21' is 52^ 30V
4. o^ 53' = 53^ 23' ; and fo for other Parts of Time j
confequently *
The Reafonof all which is evident, from Art. 5.
hereof.
16. Thefe Logiftical Logarithms may in like man-
ner be rendered applicable to Computations of Money ^
Weights, Meafures, &fc. Thusfmce 60^ : 20» :: 3 : i,
therefore the Shillings in a Pound correfpond to each
3d Degree or Minute oi Motion. And again, fince
gee*" : 3600'' :: i : 3'i; thereforeto i, 2, 3, 4, fcfr.
Farthings there anfwcrs 3', 7', li', 15^ &?^. Minutes
O 2 of
too Of Shtkw\y*sand^ett^s fables, Scx^.
of Motion h whence a Table auiy be formM to fhew
the Lo^iilical Logarkhm ^r any Number of Far*
things under 960^ or one Potind* And thus yoK
may proceed to frame a Tabic to render thefe Logif-
f teal Logarithms for Motion useful for the Ounces snd
^unds in an hundred i9^eigh 1 and thtt with great
Eafe^ fince 1792, the Ntimber of Ou«ce$ in zn hun-
dred Weight Averdupoife, a. nearly half the Numbcar
3600^ the Seconds of the Table. ConfequeAtly^
the Logiftieal Logarithm of the Dwble of any Num-
ber of Ounces, is that requited for the Pmnds au4
Ounces eq^ual thereto.
17. Or, kftly, tis caf^ for any one who fliall
think it worth while, to calc?u!atc Tables of Lpgif-
tical Logarithms^ for any Species of Con\putation»
peculiar to it felf. Thus with refpcft to Money, if
the common Logarithms of all the Farthings under
96a, be fubdufked from the conftant Logarithm of
960^ the Remainders wiH be the Logifticat Loga-
' ifithms of the Farthings in a Pound. The common
Logarithms of all the Nwnibers of Ounces under 1 792
fubftra<5ted from the conftant L-ogarkhm of the faid
Number 1792* will kave the l^gifiical Logarithms
of all the Ounces in an hundred Weight. And thus
you proceed to make Logifiical Logarithms of any
Kind you plcafe, or may have occaiion for. Which
is too eafy a matter to require an Exampfe* befidea
thpfe above,. Art. 7, and 8. .
if3m
LO-
LOGARITHMOLOGY.
PART IL
i.,i-.w»ipa -t - ■ I ■ i*^ '" "''^'
TZ^Praxis ©/'Logarithms, Common
and Logifiical :
• '
With //J AppLicATio«r to Fuhar and Duo^
decimalJrithmetiCy Plain anaSpbericalTri^
gonomefry^ Navigation^ Menfuration ofSuper--
jicies and Solids y Gauging^ Timbtr-'Meafure^
Agronomy ^ perform" d Numerically andlnjiru^
mentally.
ClfeclJSiilJ&fed^^
CHAP. L
Cftbe Rules ofADDiTion^ Scjbstraction,
Multiplication, and Division of the
laciicCS ^XoOARITHMS.
/\ prcmifed, the Ruks for a PraSical U/e and
JljL Management of thofe artijkial Numbers
vill from thence be eafily underftood, and the Ra^
tionale of every Operation be apparent to the intelli-
gent Reader. Before we proceed to the Ufe of liO-
cajithois as aralied to common Arithmetics &c. wis
^ ' ^ ' muft
102 Addition of the
jnuft firft confider {omt previous Rules and Methods
which regard the due ordering and working thofe
Nqn^bers themfelvesy on account of the Indices,
which admit of divers particukr Cafes in the Rules of
4ddition^ SubJiraSiony Multiplication^ and Divijion :
which therefore muft be exentplified and illujlratedzs
in the Sequel of this Chapter,
Addition ^/Looarithmst.
2. As in all Species of Arithmetic the ^r&funda^
mental Rule is Addition^ fo in this of Logarithms,
the Rules which require ^\s primary Operation ooxoc
firft to be coniiderM^ and they are as follow^
{Note^ I ftiall in this Pls^ce call the Indices of Loga-
rithms of wA^/^ iV«/»^^r/, Integral Indices i and
thofe of the Logarirhn^s of Decimal or Fra£ito^
nal Numbers, Decimal Indices.)
Ruk I. If the Indices be htb Jnlegralj add them
together for the Sum required.
Rule IL If the Indices are hothDecimal^ add them
as before ; and obferve, (i.) if the Sum be above or
juji ip or ICO (=tA ; fee Chap. IV. Theory) caft
away lo or loo. (2.) If the Sum be under 10 or
100 (=tA) add 10 or 100 thereto j and both the
Sum in the latter Cafe and the Remainder in tht for-
mer^ will be Decimal,
Rule III. If the Indices be of diffkrent Kinds, vi%.
Integral with Decimal^ the Sum^ if under 10 or 100,
is Decimal \ if juJl 10 or 100, or above^ caft away
10 or 100, the Remainder is Integral.
Rule IV. In cafe the Sum of two or more decimal
Indices be Jefs than io=tA, the beft way will be lo
ufc the larger decimal Indices y where tA=ioo ; and
then their Sum will be greater than 100, and fo the
Reafon of the Operation will be more evident.
3. All tbefe Rules re^te to the Indices of the Lo-
garithms only, arid are exempjified as/ollow :
Ex-
Chap. I. J«<//V« ef Logarithms. 103
Examp.I. To 3-513217 Exam. II. 2.317227
add 2.303196 . 0.850891
Rule I. Sum = 5-816413 Sum = 3^168118
Exam. IIL To .9.849235 Exam. IV. .97.237406
add .7.786822 .95.072607
^^L ] Sum= .7-636057 Sum = .92.310013
Exam. V. To .4.273760 Exam. VI. .62.346174
add .3.067247 .21.300725
% ".■ } S. = .17-341007 Sum =.183^646899
Exam.VII. To. 6.372458 Exam. VIII. .88.426703
add 2.673842 5.268402
Rule III.Sum=.9.04630o Sum = .93.695105
Exam. IX. To 5.206737 Exam. X. 8.426735
add .8.312046 .92.105374
Rule III. Sum=3.5i8783 Sum= 0.532109
*d*i $ .2.070346
II.
Part z.
Rule II. ] „ „, o
,.J S. = . 27.586181
r Or rather thus, per Rub IV.
Ex.XIi. .94.203106
.91.3127^-9
.92.070346
.77.586181
»Ti8 poffibk thefetwo laft Examples tnayfeem fome-
what oifcure •, but the Reafen and 7'rutb of each will
appear, if they are wrought at twice, as (oOfrfrs :
» To
104 Addition of the
To .4.203106 And -.94.203105
add .1.312729 .91.312729
Part a. J S. = .15.515835 Parti. J S.= .85.515835
to which add .2.070346 .9*2.070346
Adell. 1 ^ -7-— Fule II. \ ^
iyt».XS. =.27.586181 Part I. J S. = . 77.58618 1
' ■ H I ' I
FiXMn wbcAce •appears the Reaiba why» in thefe
Cafes, the larier Indicts are preferable to thtfrnaller
mus.
SuBsra ACT ION i^ Looa r i thmts.
^ Tn thfs SuhJlraSm of /;^^V^j the following
RuUs are to be obiervM, viz.
Rule!. If they are both J}i/irj^/, and the higher
one ^ greater^ tYit Remainder m\\}U Integral. But if
lbs i(Mtfr one be the gre^ier^ then add lOto the higher
oiie« md j|i#rtfff ^ the Remaini$r will be DerimaL
Ruk U. If both Indices are Ikeimal \ and the
iBi^i&^ be the greater 5 the Renuiinderh Integral ; if
not, add 10 or ico to the i&if*er one, and iiibduft
the lower^ the Remainder mil be Decimat.
Rule ni. If the Indices are pf different ibrts, ^va:.
one Integral the other Decitnai j then if the Af^iiw^ be
Integral^ add 10 or 160 to it, and iubftraft 1 the
Rmmnder wUl be Ii^^gral. But if the i^ig'i&f r be D^*
fiwr^ aod thci/fM^^r, the R^m^indfr will be Dra*
nufl ; illeffer^ the /^wy^r /|Mf^/^ muft be ufed.
5* The Rules alfo ate eafily dmy'd fipm the fore^
mtng fbe^^ and are ill«^iattd by the fpUowing
£xan4>Ies.
Exam.I.From 5*8i64i3 Ex. U. 3.168118
Suhjlraft 3.513217 2.317227
Rule I. Rem. = 2.30^196 Rem. = 0.850891
Chap. I. InSces Logarithms." io^
Ex. HI. From 3- 5 1321 7 Ex. IV. 0.850891
, fub. 5.816413 3.168 1 18
RuleL Rem. =,7.696804 Rem. = 7.682773
Ex. V. From .7.503617 Ex. VI. .94.420345
fub. .3.467306 ''9^^^73^57
RijleILRem.=4.0263ii Rem. = f.746888
Ex.VII. From .3.467306 Ex. VIII. .92.673457
fub. .7.503617 .94.420345
RuleII.Rcm.=.5.963689 Rem. = .98.2531 12
.
Ex. IX. From 5.81 641 3 Ex.X. 6.2067347
fub. .8.132700 .94.1535012
*i I I i
RalellL Re.= 7.683713 Rcm.= 12.0532335
Ex. XI. From .8.132700 Ex. XII* .94.278769
fub. 5.8 1 641 3 6.165348
RqIcIII.1 -~ RalcIII.l
Part2. 5Re.= .2.3i6287 Part2. JR.=.88.II342I
Multiplication ^Logarithms.
6. In Chap. IV. Art. 19th and 20th of ihtTleory^
the Rules for multiplying the Indices of Logarithms
(of Pure Fra£iions efpecially) are demonftratcd j and
are,
^ RuU I. If the Indeoe he Integral^ multiply as ufual ;
the Produa (hall be Integral.
RulelL If the Index be DeciihaU make the Lo-
garithm oi Unity ^ ortA=:ioo, then (hall the Index
P be
io6 ' Multiplication of the
be of the larger Sorf, wliich in this Cafe, will be more
convenieftt forUfe ; all4 theti accbrdingas yoil muU
tiply by 1, 3, 4, 5, 6, 6fr. you inuft rcjcft loo^ aoo,
306, 400, 506, (^c. from the PrcduH^ the Remtun^
der thereof will be Decimal
Ex. L Mult. 3.420673 Ex. II. 5.700672
br 2 ^
Rulcl. P/od. = 6.841346 Produfts 22.8o2l68»
■ ■ I 111 ■■•j 1
Ex.III.Mult ,96.130126 Ex. IV. .91.034106
by 2 i
Rulell. Pr.= .92.260252 Prod.s: .73.102318
Ex. V* Mult. .84.034121 Ex. VL .70.06105a
by 4 • ^ £
Rule II. Pr.= .36.136484 Prod. = .SO.gc^S^S©
In this laft Example, the Index .70. x 5=350 ; but
iince from 350 you cannot rcjeA 40C>» as p^^KuIe II %
therefore it muft be 400 — 350=. 50. which fub-
dufted from 99, leaves 49 \ which ihews that 49
Cyphers above 100, that is, 149 Cyphers are to be
prefixed \ and this you a re to underfetnd in all Cafes
where xL is lefs than af~ixtA. Sec Chap. IV.
Art.20th,of TZ^^^ry. Or thus, in general lettA = 10
or 1 00 ; then when ^L is greater than x — i x t A, then
It will be ^L — X — 1 x tA — 9 or 99, is equal to the
Number of Cyphers to be prefixM. But if ^.^i x tA
is greater than xL, then it will be x-^i xtA — xh
—9 or 99, =5= to the (aid Number of Cyphers.
Df.
i
Chap. L Lidices iy Logarithms. 107
Division ^I^oGARiTHMS.
7, This is but the Rev^r/e of the foregoing Ope-
ration ; and the Rule for decimal Indices the reverie
to that 5 which is alfo derived from Chap. IV, Art.
21ft, and 2 2d of the ^eory^ which iee.
Rule I. If the Index be Integral^ divide as ufual ;
and the ^otient-Index will be Integrals
Rule II. If the Index be Decimal^ \xk iht larger
Sort ; and then adding to the faid Index 100^ 200,
300, 6?r. divide by 2, 3, 4, 6?^. the Quotient- A^
will h^ Decimal.
Ex. I. Divide 6.841348 Ex. II. 22.802688
by 2 4
Rule I. Quot.=:3.42o673 Quotient = 5.700672
^. m. PIv. .92.26Q252 Ex. IV, .83.102300
, by 2 3
RuleII.Qu.=.96.i30i26 Quot. =.94.367433
Ex.V. Divide .36. 1 36484 Ex. yi. .30.305250
by 4 B
R»^c|I.QjJi=.84,034i2l quot. ::^ .86.061050
^
p2 CHAP.
jlo$ Mukiplicatm
OOOOCSOOOOOCSO f^ OCSOOOOOOOOCQ
C H A p. IL
QTMuLTiPLicATiON ^»^ Division of
Whole Numbers and Decimals^ I^o-
OARITHMS.
J. T TAving before fliewn the Method of finding
JlX the Logarithms of all kinds of Numbers^
both Integers and Decimals^ and alio of fitting and
adjufting proper Indices thereto ; and in the forego-
ing Chapter, the Arithmetical Management thereof
in ali Varieties : I fhall now apply the Ufe of thofe
excellent Numbers in the Rules of Arithmetic ; and
firft in the Multiplication and Divijion offFhole Num^
iersznd Decimals.
2, From the foregoing Theory (fee Chap. I. Art.
xo; and Chap. III. Art. 12.) we obtain this eafy
and obvious Rule for the Multiplication of all kind
, of Nijmhers by Logarithms^
f To the Logarithm of the Multiplicand^
viz. < Add the Logarithm of the Multiplier ;
' "^ The Sum is the Logarithm of the ProduSI.
Examples ^Integers.
Logarithms.
3. Example I. Multiply. ... 12 =s 1.079 181
by 8 =! 0.903090
Produift . • . ^ ^6 zn 1.982271
Chap. It '^Logarithms. jtof
Logarithms.
Pxamplc n. Multiply ...... 127 aqc ^.103804
by 12 =r 1.079181
Pfodud. .... 1524 = 3.182985
Example III. Multiply ..... 526 =ss 2.720986
by 100 =3 2.booooQ .
Produft .... 52600 r= 4.720986
Example IV. Multiply .... 9876 t= 3.994531
by , . . . 517 = 2.713496
Produft . . 5105892 c= 6.708071
Example V. Multiply . . . 9^7 600 » 5.994581
by ..... . 517000 = 57^3590
Produd: 510589200000 ==11.708071
4. Examples o/Mix^d Numbers.
Example VI. Multiply ...... 7,5 s=s 0.8750$!
by 10 = 1,000000
Produft 75 = 1.875061
Example VII. Multiply .... 124 = 1.093422
by, 3»6 ss 0.556302
l^roduft . . . 44*64 =s 1.649724
Ex-
'lit MukipUcation
^.ogarithmi.
Example TIIL Multiply. . 0,762 » .9.881955
by / 570 == 2755875
Produa • . 434,34 c= 2.63783q
Example IX. Multiply • . . • 36,5 == 1.562293
by . • • . , 0,00019 =s .6.278754
Produft .49^006935 = .7.841041
"" ■' <
Example X. Multiply 473 = .9.674861
by • • h ^ 1.803705
Prpdu<9: • . . » . 39.1 =: i.4785§6.
Example XI. Multiply ..... j? == 0.823909
by ^ ^S — •9744727
Produdt 2'>y^ == 0.5686^:^6
Example XII. Multiply . . . 2^,23' = 1.3 2 6541
by 4^^ — 1.623458
Produfl: ^00,71^ = 2.949999
5. Examples ofVunt Decimals.
Example XIII. Multiply .... ,12 = .9.079181
by ...... . ,8 = .9.903090
Produfl: .... jOgiS = .8.982271
Ex-
I
Chap. II. h Logarithms. fit
Logarithms,
Example XlV. Multiply . . ,0097 ss .97.986772
by ... • ^00021 ss .96.322219
Produft 000002037 =s .94-308991
Example XV. Multiply ,oo4^jz{ = .97.623458
by . . . ,0000^ = •95-948847
III II ■ 11 iw^ II ■ *m^mimm^mmm^mm,ami
Produft ,0000003735 «3 -93.572305
Example XVL Mult. ,00000085 i=: .93.929419
by i66Gooi2 3= .94.079181 '
I ' ' I II . •- • mtkm ...■. III! ■
Produft ,00009000000102 := .88.008600
Ex. XVII. Mult. ,00000000075 = .90,&75o6i
by ,000006 = .94.778151
H I I . ■ I M I 1— — »— .1.— i— ..-■■■—^—i^ili^M-^
Produft ,0000000000000045 as .85.653212
6. DivisioK ly Logarithms.
In the fame Part of the neory reftrr'd to (Art. 2.)
for th^ Rule by which Multiplicatiim is performed by
Logarithms^ you will likewifefind the Demonftmtion.
of the following Rule of D/vi/fwi ofNumbcrt by
Logarithms \
rFrom the Lorarithm of the Dividend^
. \subftraft theLogirithinofthePivifor;
w2.<Yj^g Remainder is the Logarithm of th«
L §u9ti<nt.
Examples of Ik TJ^GZKs.
Logarithm!.
. y. Example I. Divide . . . /. g6 =s 1.982271
by 12 = 1.079181
Quotient . . . • 8 = 0.903090
Example II. , Divide 1524 = 3.182985
by . . • 172 a=5 2.103804
Quotient 12 ess 1.079181
Example IIL Divide 52600 ae 4.7209S6
by 526 = 2.720986
■II I I h— .111. ■
Quotient . . . • 100 = 2.000000 •
Example IV. Divide, . . . 5105892 = 6:708071
by 5^7 "^ 2.713490
Quotient • • . 9876 = 3.994581
Exam.V. Divide 510589200000 3= 11. 708071
by . . 987600 =- 599458 1
Quotient 517000 ss 571349^
8, Examples in Mix*d Numbers.
Example VI. Divide ....... .75 =» 1.875661
by 7»5 .?== Q.875061
■ I ■ ■ wit 11 ■
Quotient • • . . • 10 ■= 1. 000000
1 Ex-
Chap. II. ^LobAftiTHM^; irj
^ Logarithms.
Example VII. Divide ..... 44>64 = 1.649724
by ...... .. 12,4 = 1.09342a
Quotient • ; . 3,6 = 0.556302
Example VIII. Divide 434>34 = 2.637830
by S7^ = ^'7^5^75.
Quotient. . ,;^62 == .9.881955
Example IX. Divide . . . ,006935 = .7.841046
by 36,5 = 1.56229 J ^
Quotient ,00019 ^ -6.278753
Example X, Divide ". . . 30,1 = 1.478566
by ...-.,... . §^ = 1.803705
tjuotient ..... *473 == .9.674861
Example Xii Divide ....... 3^,7^2$ = 0.568636
by ...;,.. ;_ ^ = 0.82390^
Quotient / =± -S-IAAT^J,
Example Xil. Divide. . . ^60,71^ = 2-949999
by ..... . 4i^P' ^ 1.623458
tiuotient '. , %h'^% = \Z}^S^\.
ti4 Dmfioa
9* Examples in Pure Decimals.
Logarithms.
Example XIII. Divide. . . . ,096 = .8.982271
by i ..... ♦ ,12 = .9.079181
Quotient .... ,8 = .9.903090
Exam XIV. Dmde ,000002037 =a: .94.308991.
by ... . ,0097 =3 .97.986772
Quotient ,00021 =: .96.322219
Ex.XV. Divide . . ,0000003735 ~ .93.572305
by 0000^ =s .95.948847
^Quotient ^>oo4?f0 = -97.623458
Ex,XVI. Div. ,00000000000102 =2: .88.008600
by . . ,00000085 := .93.929419
■ i III ■— I I I ' ■■
Quotient ,0000012 s= .94.079181
Ex.XVII. DIv. ,0000600000000045 =.85.6532 1 2
by.. ' »ooooo6 ^.94.778 151
Quotient ,00000000075 =.90.875061
io. I thiAk thefe Examples in the MuUiplication
and Divifton oi Numbers hyLogarithmSy afe fufficient
to m^xxxd: ?Ltij docible Genius in his Praftice herein ;
and as the latter are but the Converfe of the former,
fo they mutually illuftrate and prove the Truth of
«ach ether refpeftively^
^ CHAP,
viz.i
ChzpJJLRatfingPowirityLoQAiBLiTnMS. 115
chjlP. hi.
OfraifingYovi'g'BiSy and the Extract lov tf
Roots by Logarithms.
I, "ITtROM theTheoiy of Lc^rithms (Chap. I.
J|7 Art. 12. and Chap. III. Art. 13O wc have
an evident Rule for the Involution of jffm^^Sj or
raifing them to any propofed Power by means of Lo-
garithms \ which is this,
f Multiply the Logariibm of the given Number
by the Index of the Power, viz. 2, 3, 4, $%
(^c. the ProJuff {hall be the Logarithm pf
the Power, viz. the Square^ Qube^ Biqua-
drate^ Surfolid^ &c^ Power of the iaid giiien
Number.
Examples in iNyoiuTirfM.
2. Example L What is the Square of the Num-
ber 32?
Multiply the Logarithm of . . . . • 32=i.505i5a
by the Index of the Power . * . . . 2
TheProd. isthe Log. of the Square 1024=3.010300
Ex- II. Required the Square of 3,2=0.505150
Multiply by • ^
The Prod, is the Log. of theSquare 10,24=1.010300
En. III. Required the Square of . . ^^2:^.9.5051 50,
Multiply by . . ^ »
:{'hsPi;o<!luai».th^A;ifwer . . ,i024=.9.oio30o.
Xi6 OfraifingFw^eri^
Example IV. Required the feveral Po^/frcrs of the
Number 1.05 to the Surfolid?
1. The Logarithm of 1.05=^.021189
Multiply bjr .•..,.. . 2
The Pxodua is the % Jr^ . . . 1,1025=0.042378
2. Multiply the Logarithm of 1.05=0.021189
by 3
' ';|phe Produft is the Cube 1,157625=0.063562
3. Multiply the Logarithnd of 1.05=0.021189
by ...:.. 4
Frodud: is the Biquadratt I.ai65o6z5=:0.o84756
4. Laftly, Multiply the feme . . 1.05=10.021 189
by ^s^
' VroduAistht Surfolid 1.2773315625=0.105945
Example V. Reqi^ired tjhe Jfirfilid ]poiyer of the
Number ,0006 }
Multiply the Logarithm pf 30096=.96.77i8 151
By the Index of the Power 5
The SurfoL ,00000000000000007776=. 8 3,. S^qjss
Example VL What is the CuboXuheVo^tx of ,08 ?
Multiply the Logarithm of. . ,o8=.9 8.903090
by the Index of the Power ... 6
The Cuh'CuieFov9tt ,000000262 144zz.93.41 8540
Example VII. What i§thc 57th Powenofthe Num-
ber 399 ?
Multiply
j
Cliap. III. 6y lyOGARiTHMs. 1 17
Multiply the Logarithm of . . . 599=^.9.995635
%y th? Index of the J^ower ... gj
69969445
49978175
The 57th Power IS . • . ,56389, &c.=.9.75i 195
3, There is another way of raiBng the Powers of
Decimal Numbers by Logarithms, and it is thus 5
r Multiply the Aritbtnetical Complement of the
1 Logarithm of the given Fraftion by the
viz. < Index of th^ Power, the Arithmetical Com-
I plement of the Produdi: is the Logarithm of
t. the Power fought. •
Apd this in many Cafes, (as when the Xndex pf the
Power is a mix*d Number or pure Decimal) will be
fpund moft certain and ready. Thus in the laft £xt
ample this way ;
Exarpple VIIL What is the 57th Power of the
Number ,99 ?
The Logarithm of 399=-?-995635
The Arithmetical Complement 0.064365
whiqh multiply by . . ^ gy
■ . ^
21825
The Produfl: . . . 0.248805
The Arithmet. Comp, is 1 ^r^o^ «,^ ^^ ^
the Log. of the Power J '56389> ^^.=^9^751^5:
' ^xamplelX. What i« th^lth or 25th Power of ,z ?
Tha
JiJ J^:ictra£fion of RB0ts
The Logarithm of . . . ^aszi.p.goiojo
The Arithmetical Ccmp. thereof ,0.698970
which multiply by the Index ,25
3494850
1397940
TheProduft . . . 0,17474250
The Afithmet. Comp. is j ,^0 o
the Log. of the Power \ '^^^74, 8{c. =.9.825258
Example X. What is the 6,25th Ppwer of ,0032 ?
The Logarithm of . . . >oo32=:. 7.505 150
The Arithmetical Complement 2.494850
which multiply by . . .' ^ 6.25
•
The Produft is . . . 15.5928125
The Arithmetical Complcm. of which is .84.4071875
And the Number anfwering thereto, viz.
3O0000000000000025538 is the 6.25th Power of
5O032.
4* EvotuTioN^r Extraction <?f Roots
^j' Logarithms.
This is done by a Rule, the converfe of that fovlnvo^
lution^ in Art. ift 5
{JDivide the Logarithm of the Power by the
Index of the Root, the Quotient (hall bQ
the Logarithm of the Rootlbught.
Examples inRvohvr 10 u.
Example I. What is the Square Root of 1024 ?
Divide the Logarithm of . , . 1 024=3. 01030Q.
By the Index of the Root 2
The Qudt.is the Log .of the fquare Root32= 1.505 1 50
Chap. III. by Logarithms. i 19
Example 11. Required the Cube R$ot of 1,157626 ?
Divide the Logarithm of 1,157625=0.063567
By the Index of the Root 3
The Qu. is the Log. of theCtf^^ R. 1,05=0.021 189
Example III. What Is the furfoUd Root of the Power
,00000000000000007776 ?
The Logarithm, thereof is . . . *^i'^9oy5S
which divide by Ae Index 5
Example IV. What is the" Guho-Cuhe Root of the
Power ,000000262 144 ?
The Log. of . . ,000000262 I44c=.93.4i8540
"Which divide by the Index 6
The Root fought is . . . ,o8«=.98.903090
Example V. What is the 57th Root of the Power
The Logarithm of . . . ,56389> &c. .9.751195
which divide by . . . 57
The Root required is . . . >99='9*995^35
5. Another different way to extt»ft the Root of
Decimal Numbers is the convcrfe of that in Art. 3d,
hereof.
r Divide the Arithmetical Complement of the Lo-
I ;garithm of the Decimal given by the Index
viz. \ of the Root required, the Arithmetical Com-
i plemcntofthe Quotieut is the Logarithm
L of the Root fought. •
Ex-
1.2b EkfjraSliondfRobtiy &c.'
Example VI. What is the ,25th Root of the Po wet
,66874, &c. ?
The Logarithm thereof is . . . .9.825258
The Arithmetical Complement is 0.174^42
which divide by . . . ^25
The Quotient . . , 0.698970
The Arith. Comp. is the Log. 7 ^ ^^ ^^ , ^ .^
of the Root "^ ^}- • •2=-9-30i030
Example VIL Required the 6.25th Root of the
Power ,00000000000000025538 ?
The Logarithm thereof is . . . .84.4671 8 Jr
The Arith. Comp. of which is 1 5.5928 1 3
which divide by . ; • 6.25
The Quotient is . . . 2.494850
the Arith. Complem. thereof is . . . .7.505i50
the Logarithm of ,0032 the Root fought.
Example VIIL What is the CuboXube or 6th Jloot
ofthcPower ,1 ?
The Logarithm of . . • ,i=.9.ooooo6
the Arith. Comp. thereof . . •^•999999
which divide by the Index 6 .
The Quotient ... 0.166666
The Arith. Comp. is the 1 ^o,^^ o, o
Log. of the R^ot i .68129, &(;, =.9,833333
6. Thus you fee the great Ufe of Logarithms iri
cxtradliog the Roots of a given Power, which tho* a
thing fo very difficult by the Rules of common Arith-
metic, is yet render*d moft eafily prafticable by this
excellent Art ; yea 'tis eafy to make it appear, that
the ExtraSion of Roots is not only moft expeditioufly
performed, but hath a more univcrfal Perfeftion in
this Method, than in any other.
chap;
chap* IV. Faridusttukstf Proportion, ?21
CHAP.' IV.
Of the various Rulis g/'lPROPORtioN, and of
finding Mean Proportionals ^y Loga-
RITHMSk
* • 1? R O ^ *^* ^wry *t>* evident, that the golden
J[7 ^«^^ Of ^'' of Proportionalsy is wrought
in Logarithms by only the Addition and Suhftra&ien
of the logarithms of the Terms of the Proportion.
And {f the Proportion be direA^ the Rule is thus )
r Add the Lx)garithms rfthe fecond and third
. . \ Terms, from that Sm fubftrad the Loga-
^'^•] rithm oiihajirfti the Remainder is the
£ Logarithm of the fourth fought.
Examples in the GoldAn Rule Diteff,
Example Iv Ifit2Pbundscoft il.iss.$d, what will
173 Pounds coft? . '
The Logarithm of . /. I2=:i.679i8i
f o the Log. of i/. t^s. 9 i.=i,7875=so.252246
Add theLo^thm of 173 =2.238 046
theSum..... 2.490292
Subfttaa the &ft, there refnains 2^,77 /. =i.4inii
Wherefore the Anfweris 25,77 /.=25/. *5^- 4<^.i»
2. But fince if ,you divide by any Number, or
multiply by its Reciprocal, the Eflfeft is the feme j
andalfo fince the Arithmetical Complement of any
Number, is but theL©garithm of the Reciprocal oi
122- Of the varioui kuks
that Number; therefore it follows, that where the
Sf/^ra&i(m of a Logarithm is required in any Ope-
ration, if you take die Arhbmettcd Cemflement eS
that Logarithm, the^hole may be performed by
Addition only. Thos m the foregoing Example.
{the Ar. Comp. of the Log, of 1 2.=. 8.9208 19
the Log. of •^. 1,7875/.= 0.252246 •
the Log- of ^ 1732x2.238046
The Sum is the Anfwer • • • . 25,77 '•=141 ^ m
the fame as before*
Example II. If 2C. iq. 21U i^4fz.CQ&5t. ijs. tiX
what will 31 C. zq. 26 1. 150%. coft ?
Then 2 C. iq. mL i4^.:s:a.4453A.Coiit. .9.61 1668
The Log. of $1. lys. Sd,i^ 5.8844 = 0.769702
« 111 II
The 5«iw IS the Logarithm of 76,38 i/.=:i. 882984
Therefore j^^iiL^xfuSL js. yd.i is the Anfwer.
3. . 0//&r Role of Three /»wf/Jr.
In this Caie you mufl take the Arithmetital Conh
flement of the tbifd TerW, and add it with the Lo*
gu-ithms of the other two asJbefore \ ib ihall the
Sum be the Log^ithm of the Anfwer.
Example* Suppofe a Field feeds 18 Horfes for 7 v
Weeks, how long will it feled 42, at that rate >
{the Logarithm of 18=1.255272
the Lo^rithm^of « 7=0.845098-
the Arim.Comp. of the Log. of 42=.8.37675 1
The Sum is the Lc^arithmof Anfu^r 330.477 12 1
4.:
f
Chap. IV, ^ Prfifartim. » J23
4. Oftbi dotthle Rule f^tbru^ cr Rule pf
Five Numbers.
As in QjiefKohs of this iQrt» diete axe always tbue
idndiiwHaToc Jufpofed Terms ; the £rft of which is the
principal Caufe of Gain^ Lofs^ ABitm^ &c the fecmd
^ denotes the 7/i»r, Difiance^ &rc. and the third is the
Gain^ Lofsy or AEtitm^ &c. So kt thefe three Terms
be denoted bjr the C34>ital8 P, T, G. Alfo there are
three other Terms (fimilar to the three former) which
make the Qjieftion to be lefolv'd % and let thefe be
wprefented by the fmajl Letters pt t, g. Two of which
are always given, and the other is fought. But fince
P.-Girpi-^^ and again, fincc T:^::t:gi
therefore Tg = ^ , and confcquently PTg=tpG ;
from which general Theorem we am eaftljr find p» t;
or g. Thus, 1 i^ = Pi and W.^^izxAHl.
J~ as g. The Contrivance of thefe excellent The-
orems weowe to the fete Mr. Wari^ cXCheJtir.
5. I fliall exemplify Queftimski this Rule by Ex-
.amples, as follow.
Example I. If 100 /. ini 2 Months gain 6 /. what will
350/. gain in 9 Months ?
Here P=ioo, T«:i2, Gx=s6 j alfo p?=350f t=s9,
to find g.
f the Logarithm of, ..•.., G=;=6=:0.778i5i
Add< the Logarithm of p=3 50=2.544068
(^the Logarithm of t==9=?o.954242
The Sum is the Inqgantbm of « • . ^Gpfc=4-2 7^461
R 2 Add
i 24 Of finding Mean Proportionals
^•,c the Logarithm of . . • . . P==ipoi=:2.oooQOp
I the Logarithm of T=i2=i.©79i8i
The Sum is the Logarithm of pT=;3.o79i8i
Then from the Logarithm of ... • Gpt=4.27&46i
Subdudl the LogOTthm of PT=3.o79i»i
There rem. the Log, of ^5=g== 15,75 =1.197280
Wherefore the Apfwer is i^.^sl^i^l 15/.
6. Examxple IL If .36 Bufheb wijl ferve 24 Horfes
48 Days, how long will 126 Buftiels ferye
96 Horfes ?
Here P=s24, T=48, Gc»3(>; alfQp=96, and
• g=i26i to find t. -
f the Logarithm of P=24=:i .3 802 1 1
Add^ the Logarithm of T=4fc=i.68i24i
l^the Logarithm of g==i26=?. 100370
The Sum is the Log, of . - PTg=r5-j6i822 frpm'j,
i. J. C the Logarithm of G==36=|. 556302 '.
■ c the Logarithmi of pc=96=i. 982271 |
The Sumi is the Log, of ... . Gp=:3. 538573 fubf.^
Thefefo|« Ae Lag. of H^ = t = 42 = 1.623249
ThcAnfwcr, viz. 42Wec^s.
7. Example III. At the I^tc of 4s/. per Cent, per .
Am. what Principal will produce' 35/. 15^.
in 7,5 Months ?
Here |*=|oo, Tsb:i2, Gssi4,5 j alfo tssr7.5, and
6=25-7^ « to find p.
J
Chap* IV. by Logarithms. %z$
{(be Logarithm of P=:ioQ=2,ooooop
the Logarithm of T=i %^\ .079 1 8 1
the Logarithm of g=35,75=i.553^76
ThcSum is the Log. of . . . PTg=4.632457 fromy
- , J 5 *^* Logarithm of 0=4.5=0.653^2 I
^^ \ the Logarithm of 1=7.5=0.8 75061 j
The Sum is the Log. of 01=1.528273 fubcj;
Whence the Log. of^?=p=i27i ,1 I4=3.i04i84«
Confequentiy, I27i,ii4/.=i27x/. 2j. 3^4 is the
Anfwer.
NotCy Thcfe Theorems give the Anfwer ahTplutetyr
without regarding whether the Proportion be DireS
or InvirfCj or both together, as in Art. 6. Exam. 1I«
8. Of the Method of finding Mean Proportionals,
In* order that a clear Notion of this moft uftful
problem may be had, I (hall premiie the followbg
things.
1. BtVHtexi two fquare Numbers AK and BB, there
will fall but one Mean Proportional AB ; that is^
A*: ABS:: AB:B*. SetEucl.i. ii.:
2. Between two Cubic Numbers A^ and B*, there
will fall two Mean Proportionals A*B and AB* ; that
is. Ah A^3;: AB* : B». See Eucl. 8. 12.
3. Between two Biquadrate Numbers A^ and B\
there will fall three Mean Proportionals A'B, A*B\
and AB» •, that is, A-^ : A«B :: A^B : A*B* :: A*B* :
AB» :: AB^ : Bl
4. Again, between two furfolid Numbers A^ and
B», there will fall four Means^ viz. A^^B, A^B%
A*B% and AB^j that is. A*, A^B, A'B\ A*BS
AB% BS will be Proportionals. And thus the
I Number
ri26 Of finding Mean Freportionah
Number of Mean Proportionals will tc always le/s hf
one than the Index of the Fowcr of the 0ven Ex-
tremes.
9* But the comrnm Ratio of all fuch Seriea 13 \
ForAAxI =AB; tndABx \ «BR
AndA«x?.=A*Bian(JA*Bx-? =AB*; and
AB* X 1- sc:B*. and fo in the others. Here I have
A
fuppofcd A tobe the Jcaft Number, and B the greateft,
and the Scries to begin firom A*, A*, ^e. But if B
be lefs than A, and the Series begin from B*, BS fcf <•.
then the Raeio of the Scries will be •g •
Now| isRootof ^,,5^, 5>^^'*^^^-
fbre from thefe Prcmifes well underftood, 'tis ca{y to
(:onceive the Reafon of the following Rule for fold-
ing Mean Proportional?, vi%.
"SubftraA the Logarithm of rhcfe^j^firwfrom
the Logarithm of the greatefiy and divide
the Remainder by a l^urnb^ greater by one
than the Number oi Means defircd ) then
Ilule^ add the Quotient to the Logarithm of the
Icaft Term {or fubftraft it frofh the Loga-
rithm of the greateft) continually, and it
will give the Logarithm^ of ^ tl^e ^Seavi
Prop4iriumals defired.
10. Example L Let two Mean Proportbnals b^
fought between 8 (=A^) and 28 (=;; B\)
♦i' ■
The
Chap. IV. bf Logarithms. iny^
The Logarithm of B*sss28aBX.44f7i5S
The Logarithm of • • ^ . • . * A^:te8=^o.903O9o
The Difference Is ^, so.544068
Which divide by 3
B
The Quotient is . . • "J =0.181356
To which add the Log. of . , . A^s8sBO.9a309o
■^.ft^ th. Log. of J A'B-»,.^l.o«444«
To whkhaddagain -^ sso.iSijfS.
\
The Log. of the 2d Mean AB*si8»44=si.265loz
■ -,i
Wherefore thePropor- > 8 : 12,14 :: 18.44 • 28
tionakaie S A* : A*B :: AB* : B*.
XI. Example II. Between x 6 and 64 find £ve
Mean Proportional^.
The Logarithm oif B*=r54»i.8o6i$c^
Subdudthe Logarithm of. . A^ssx 6s 1.204 120.
B^
There remains ^ . . ^ =:p.6o2o6t>
B
J of whichis -^^ =0.100343
To which add the Log. of A*s;:i6=;i.t04i2o
*l'T.S^^- "'l A.B=.o..5b...3a^3
To which add again ^ :so.ioo343
The Log. of the ad Mean A*B*ia:25.398tti.4048pfr
And
J28 Simpte Mereji
And thus you produce the Logarithms of^
{3d Mean ; . A*B»=32=i.505r5<j
4th Mean A*B*=40.3i7=:i.6o5493
5thMcan *. . • V . . . AB«=50,796=i.^05836
-«Ei-
\
The Series tlierefbre is this,
15.20,158/25,398. 32v 40,317- 50»796- i4-
A^ A>B. A4B\ A^B\ A^B*. A'B. %\
t2. If it were required to find 3^4 Mtan Propor-
tionals between o and i .06 ; or o and i .05 ; or o and
1,04, &fr. then A***=o, and fi^**=i,o6i 1.05,
Cfr. and fo -^ ^^^s/ZoSl otVlioJ, tff*
Wherefore if you multiply the Logarithm of B, by
2> 3» 4» 5» &c. to 365, you will thereby obtain
the Mean Proportionals rtquittd. And thefe will be
the fcveral ^mounts of i /. and its Intereft^ for each
Dtfy of the Year, at the Rates of 6, 5, 4/. &c. per
Cent, per Annum^ Compound Intereji. But more will
be (aid of this hereafter.
#
C H A P. V.
Simple Interest iy LogaJhtHMs*
I. Ti ^ Y Defign being only to acquaint the Reader
iVJL ^^^h the Theory and pra£lical Vfes of Lo-
garithms^ and not to treat of the Theory of any other
Art or Branch of Mathematical Science ; it will be
fufficient for me barely to mention the Theorems or
Rules^ on which the divers Parts of Learning (I
(hall treat of; depend, and fhew how they are moft
conveniently wrote by Logarithms.
Of
0/S I MP LE IkTZRSST.
2. 1 have more than once ferv'd myfclf with thofe
excellent Theorems of Intereft contrived by the late
ingenious Mr. fFardi and ihall once again makis
them fubfervient to my Deiign in this Place. In
order to which
f P=any Principal or Sum put to Intereft.
J. \ R=the Ratio of the Rate^ per Cent, per Ann.
1 T=the time the Principal continues at Intereft.
I A=the Amount of the Principal and Intereft.
3. Then any three of thefe being given, the other
may be found by the following ^theorems.
Thcpr. I. TRP+P=A. Theor. II. .j^ = P.
Theor. HI. —^ = R. Theor. IV. ^ = T.
Queft. I. What will zysL igs. amount to in 3^
Years, at 4iL per Cent, per Annum ?
Here V=275yy5, T=3,5, R=o.045, to find A.
Theor. I.
{the Logarithm of . . . P=2 7 5. 755=2 .4405 15
the Logarithm of • • . T=s 3.5=0.544068
the Logarithm of. .. R=so.045=.8.6532i2
The Sum is the Log. of PTR=43.43i=i-637;^95
To which add P=275,75
Queft. II. What Principal or Sum being put to Ip-
tereft, will amount to, or raife 5 Stock of
- 319 ^- 2^* 7^* i" 3^ Years, at thQ rate of 4^'*
per Cent, per Annum ?
^ S Th«.
'f^6 Simpklnterefi
The Log. of the Amoum A=/. 3 1 9, 1 8 i=a .50403 6
- jjCthe Log. of the Time . . . T=3,5=o. 544068
• 1 the Log. of the Rate R=o.045=. 8 .65 3212^
The Sum is the Log. of . . TR=o.i575=.9.i9728q
Then the Log. of ... TR+i=;=x. 1575=0.063521
Subftraded from the Log. 1 ti ^^ ; *
of the Amount leaves ^ I P=275>75=2.4405i5
yrhcrefore the JV7»f/>j|} ,„^- ^-_«^^; ,^,
required is \ Uy5'75=^75l' ^5s.
Queft.III. At what Rate ^frC(f»/.&c. will 275/. 15 J,
amoant to 319/. 35; yd. iii 31 Years ?
Here A-:-P=3 1 9, 1 8 1—2 75,75=43.43 1 .
The Logarithm .... A — P=43.43 1=1. 637800.
...^t the Logarithm of . . . P=i2 75,75=2.4405'! 5
- I the Logarithip of . . . T= 3,5=0.544068
The Sum is the Logarithm of PT . , . =2.984583
* Queft. IV. In what Time will 275/. 15 jr. raife n.
Stock oif 319/. 3j. 7<;. at the Rate of 4I/.
fer Cent, per Jnnum?
■^ \ the Log. of the Ratip R=0,O45=. 8.6532 ? 2
The Sum IS the Log. of PR= 1.093727
Which fubftraft from \ a p_, ^ , ^' _, a;.^q^^
the Log. of I A~P=4M3i=i.6378oq
Thene^
chap. V. by LooAliitHMs. 231;
There will remwi the j A-^p
Log. of \ "^=^=3^5=0.544071
Therefiwc the Anfwer is 3* Ycais.
^ 4. Of Annuities, Csfr. in Arrears.
!rtJ=the Annuity^ Penjion^ or yearly Rent.
T=the TVfw^ of its Continuance iinpsiid.
^ut / R=the Raiio of the i?/?/^ of Intereji.
. I A=:the Amount of the Annuity and its tntereft;
Then the theorem f<ir finding each of thofe Parti*
tulars, are as follows.
Theor.t. I2izSJxR:+TU=A.
Theor.II. xrk— tr+zT =^*
Theor. III. yr(;f_yt) = R.
Theor. IV. ^^W^ • — i'' =*^T.
Queftioill. If 250/. ^^ar/y Rent (Penfion, €9*f.) be
fbrbom or unpaid 7 Years, what will it amount
to in that time at 61. per Cent, for each Payment
as it becomes due ?
Her6Uss250, Rss=o.o6, Ts» 7 j to find A^fer
Theor. I.
- , J 5 the Logarithm of .... 17=250=2.397946
'^'^'^i the Logarithm of . . ; . Ta«= 7=0.845098
The 5«<» is the Log. of .. TtJ=:i750ta3.24303d
add the Logarithm of . . . T=« 7mo.845098
The Sum is the Log. of TTU=i225oai=4.o88i28
S a Then
IJ2 Simple Intireji
Then the Log. of '^"— "^ =5250=3.720159
to which add the Log. of Rs=o.o6i=:.8.778 15 1
The Sum is the Log. of^^SzSJxR— 3 15=2.4983 10
To which add TUs:i75o
TheSumisHHrlH? ' ~ , ,
„ ,^,\ t 2065/. the/
xR:+TU=A= 3
Note, if thefe Payments be made
rQjMjrterly. -)^ pR, iU, and 4T.
' Half-yearly, J. « <4R, 4U, and 2T.
(Thrce^uarterly,J^ (|R, iU, and IT.
Qucft. II. What Tear-Renty Penfion, &c. being for-
bom or unpaid feven Years, will raife a Stock
of 2065/. allowing 6 per Cent per Annum,
for eadi Payment as it becomes due ?
Here A=2o65, TaB7, Rs=o.o6 j to find U, per
Theor.If.
f^^^\t\itIjo^ihmo( Tss/sb: 0.845098
I the Logarithm of . . . RssO,o6ss:.8.778i5i
The Sum is the Log. of. . . TRs=:o,42=.9.623249
to which add the Log. of T=:7-5 0.845098
The Sum is the Log. of . . TTRsa2,94= 0.468337
From which fubdu^l: . . . TR=o,42
There remdns . , TTR— TR=2.52
to which add ...... 2Ts=:i4
the Sum is TTR— TR4.2T=:i 6,523= 1.21 8010
the Lo^rithm of 2 A=4i 30^=3.615950
The
Chap. V* ^Logarithms. 135
The DiiFcrcnce of thcfc iHOgarithms is U=a25o/. sat
2.397940, which is the Annuity fought
Queft. III. If 250/. Yearly Rent, 6?f. being forbora
7 years, will alhount to 206 5 1. allowing
iiiAple Intereft for each Payment as it becomes
'^^ due, what muft the Rate of Intereft be pir
Cent, per Annum ?
-. , . C the Logarithm pf . . . . U»250aB2.39794a
t the Logarithm of . • • . Ts3s7s=s 0.845098
The Sum is the Log. of . . . UTr-i 750=8:3.243030
to which add the Lc^. of . . . Ta7ss 0.845098
ThfSum is the Log. of •TTUr=:i2250ar4.o88i28
from which fubduft .... TUs=i750
there remains TTU— TU=: 1 05005=54,02 1189
the Logarithm of . a A — 2TU=:630s=52.79934o
Subduftthe former from the 1 ^^ ^ ^^ o ^^o ^
kttcr,th«€ will remain } ^^0.06^.9. jjZi^t
Wherefore, as x/. : 0.06/. :: 100/. : 6/. the Rate re-
quired.
Queft. IV. In what time will 250/. Yearly- Rent,
raiie a Stock of 2065/. allowing 6 /.p^r Cent^
&c. for the Forbearance ef each Riymentas
it becomes due ?
Here 17=250, A=2o65, R=sio.o6, ^ — 1= ^
— i=3^>2^s=^5 to find T, ;>^r Theor. I V.
- , J c the Logarithm of ... . U=250c=:2. 397940
cthc Logarithm of . . . • R~p.o6=.8.778i5x
m ■ n *
The
134 Simple Intereji
The Sum is the Logarithm of RUi=:i 5^=1 . i y6og t
ivhich fubf. from the Log. of 2A=:4i30=s:3.6i595o
2A
there remains ^ «275>3^=^-439859
Addi *^ Logarithm of xzsaa,^'^ 1.509658
(the Logarithm of . . * . i;c=8.o83'=d.907594
XX
The Sum is the Log. of 7- =261,3605=2.417252
to which add ,..•.. Iir=275»3333
the Sum is . . ^ + tT . =536,6938=32.729724
Half that Dm. 2 ^^A - xx ^,^ ^ ^^
istheLo^ J RU + 1 =23,166^=1.364862
from Which dedu6t . . * • ix=i6,i66^
there wiU temain "^^ + ^ : ~4x;=t=7, the
Number of Years required.
5. Of the Present Worth ^Ani^uities,
Pensions, Gf^.
Here U, P, R, T, are ufed to denote the AnnuHji
prefentWorihi Rati$ of the Rate oilnterefi^ and
^ime^ as in the former Articles ; here alfo let ■^—
U — i=Af ; then the Theorems for Operation are
as follows \
Theor. I. "^^l^^fc^I 5,u=P= Prfefent Worth.
2TR+2
ppf X 2P— U,=rAnnuit
Theof.
TR+t
Theor. 11. ttR— TR+zT ^ 2P— U,=rAnnuity, 6fr.
Chap. V. b} Logarithms. 135
iP— •2TU
TheorJII. TTiTZftjirSPT =R=,Ra^o of the Rate,
Thcor. iv/^ + "J =±= ^^=T, = the Time.
Qucft. L What is the prefent Worth of 75 /. Yearly
Rent, to continue* 9 Years, at 6 fer Cent.
Sec.
Here U=75, T=9, R=o.o6, to find P, per
Theor, I.
^ ,j cthe Logarithm of T=9=o.954242
'^^"1 the Li^arithm of R=o.o6=.8.778i5i
The Sum is the Log. of . . . . TR=,54=.9.732395
Alfothe Logarithm of. . TTR=r4,86=o,686635
fubftraft TR=,54
there remains . . . , TTR~TR=4.3^
tp which add 2T=i8.
The Sum is TTR—TR+2T=2^,32=:j. 348694
the Logarithm of . . . 2TR4.2=:3.o8=o.48855i
the Difference is * =0.860143
To which add the Log. of . . . 11=75=1.875061
The 3um is the Logarithm of
" ^aTlIz"^'^ X U = P = 543,506 =2.735204
Wherefore the prefent Worth is 543/.. tos. id A.
Queft.II. What Artnuity, Penfion, 6fr. may bei
purchafed for 543/. 10 s. i^. I, to continue
9 Years, allowing to the Purchafer6p^rO»/.
ftr An. fimple Intereft ?
Here P=543,5o6 5 T;=:9; R=o,c65 to find U, per
Thcor. \h
From
136 simple Jntereji
From th«5 "Log. of TR4-i=i,54=o.i8752i
fuM^Acj TTR_TR+,r=..,3«,.348694
to the DilRrence .8.838827
add the Logarithm of 2P=io87,or2=3.0362j4
The Sum is the Logarithm of
fTll^yft+at X 2P = U =5 75 =1.875061
That is, 75/. perjnnun^ is the Amiuity, fcfr. fought.
Qucfl;.III. If 543/. 10 s. I J, J- ready Money, will
purchafe an Annuity, Leafe, 6ff. of 75/.
per AnnufHy to continue 9 Years ; ^uare the
Rate of Intcrcft per Cent. &c. ?
Here P=543,5o6; U=75 \ T=:o 5 to find R, jp^r
Theor.UI. '
From the Log, of 2? — 2TU=:262,98 8=2.41 9936
*L?g. rf} TTU-TU-2PT=4383,io8=3.64i782
the piiftrence is the Lpg. of R=o.o6=.8.778 154
Wherefore, as i/^: 0,06/. :: lOp/. : 6/. the Rate.
rc<juired.
Queft. IV. If for 543/. los. %d.il purchafe an An-
nuity, Penfion, 6fr. of 75/. per jiHnum%
^are^ how long I may enjoy it, at the Rate
of 6 per Cent. &c. Intereft r
JJcre P=543,5P6, U=75, R=o,Q$i tofindT,^fr
Theor.IV.
From the Log. of . . . !;}P=io87,oi 2=3.036234
fubduftthe Log. of 17=75=1.875061
2?
the pifF. is the Log, of. . g- =14,493=1.161173
WWch
chap. V. by Logarithms. i^f
which fubftraft from • . ^j -^;~32^33J
there remains g- — j^ — is=s;fesi7,840=i.25i395
add the Log. of 4^=446=0.649335
the Sum is the Log. of, . 4xAr=79,5664= 1.900730
to which add ^ =241,558
the Sum is ^ + *^ =3321,1244=2.506659
half which is J ^/Ip TTT
theLog.of} ^Ku +-; =17,919=1.253329
fubftrad ix=8,92
there remains v^J^ 4.^''— ix=Ts=8,999=c9,
the Years fought.
6. If the Queftionbe of Annuities, &?r. in Reverfiony
you muft find the Amount of the Purcbafe" Money to
the time of G>mmencement (together with itslntereft)
by Queft. I. Art. 3. and make that Amount the Sum
for the Purchaie ; and then proceed as in the Que*
ftions of this laft Article. Thefe are all the fund4h
mental or original Czies of ^mpie Intereji.
^
CHAP.
CHAP. VI.
CoMpauND Interest by Logarithms.
I . ^T^ H E former Qycftions o^Jimpk Inter e^ ipight
X be refolved by x\i€ ^\A€% oi vulgar Ariih*
metic \ and I have only there (hewn, they may alfo
be (and that in mwy.Ga^s, moft'convifiienfly) wrought
by Logarithms. But fa the- prefent Affair of compound
Jntcr^j the Ufe of Logarithms is ahfolufely necejfary ;
no other Method pf the Solution of Queftions in
compound Intereft being equal- to- it in- Extent and
PerfcAion. And cbnfc^jcutly the young Student in
Arithmetic is under an indiipeniible Qbli|ptioii.lx> be
acquainted with this moft excellent and uleful Branch
ot the Science.
2 . I fcall here alfo- proccicd according tq the Theo-
rems of Mr. fFard ; and therefore
r Prrthe Principal put to Inter efi, . ,
\ t=thc ^ime of its Continuance.
Put^ A=the Amount of the Principal and Inter0.
• I R=:the Amount o{ il.znd its Interefi for one
t Year, at any given liate.
Note^ you find \ lOO : io6 :: i : i,o6=:R, at SperCt.
^ R thus, J loo : 105 :: i : i,05=R, at ^per Ct.
The Amounts of i/. c i. 2. 3. 4. 5, fcfr. Years,
in fcveral Years \ R. R* R» R^ R^fcfr. Am.of i/.
Therefore Rt = the Amount of i /. in the Time t for
the Rate agreed on ; this being premifed, the Theo-
rems for Compound Intereft are as follow.
3. Thcor. I. Y'^'znh.^xh.t Amount,
Theor^
Chap. VI. ^/LoGARtTHM^'; ^3^
Theor.il. I =?= the ?rinclt)ai. ■ ■
Thcor. III. ^ = R* I — the Amouotof t/.
By thefe Theorems the feveral Cb^ftions «f Com-
pound IntenJftrare ahfw^ed sftOft expeditiouay by
the Li^aritbms in th« M»nnw folloMving. .
4. Qgeft. I. What ^1 27s/. 15^- amount toin
3! Years, at 4 1, fier Cent, per Jnnmi Com-
pound Intereft ? " •
HerePs!«275,75 4 Ra=i,04^i t:s=3,5'j ihtn^ A,
ferTbeac,: I.
The Logarithm of R»i,Q45:=:o.oi9ii6
. multiply by the Time ... .t ^ . ^5
■ The ?^. 19 the Log. of R'^i ,1665=0.066906
To which add the Ldg. Of P— 275»75^^-4405i5
TKeSumistheLog. of PR«=.^i2 1;,68==~2. 507421
.. $0 the Amoppt fought is 22 i.Li 3^s. 7d.\ which i»
m6te tJ^n thi'Amojvitlby finipre rntereffi by ij.jos.
S*(^cff,I. Alt. 3. of ^he foregoing Clis'tJta^. '
... ■ . . -^ . : -■■J , -T
Oueft. IT. WHat Principd or Sum ^Ing piV te Vfe
at 41 p& C^. eoiiipdutid jWereft, will ■
ambfint to 32 1 /. 1 3 i. 7 d. in 3I- Years ?
Here A=c32 1,68-, R=i,045i t=3'5 J to find P,
per Theor. II.
Frdm-ffieLogarithmof A=32i,68=2.50742i
Subduft thel5>garithra of R'=i.i665=ao^69o6
TheDiff.isthfiLog. of^.*=:P=a75»75?=f2-4405».5
therefore the Principal fought is 275/. 15 J.
. T 2 Q««ft-
/
1 4» Compwnd Inter efi
Qucft. III. At what rate pr Cent. &c. Compoun4
Intereft, will 275/. 15J. raifc aStock, or a-
mount to 321 /. 13 J. 7^. in 3I Years ?
Here A=32 1,68; P=275,75; t=3.5 ; to find R,
ftr Theor. III.
From the Logarithm of A=3 2 1 ,68=2 . 50742 1
Subftradtthe Log. of. . . F=275,75i=2.4405i5
The Difference is the Log. of R« 0.066906
which divide by the Time ... ^Z^S
TheProdudlistheLog. of R=i.Q45=q.oi9ii6
Then as i : 0,045 :: 100 : 4,5=4^/. pr Cent, the
Rate required.
Qucft. IV. InwhatTimewill275/. 15/. raife a Stock
of 3 2 1 /. 13 s.'ffd. at the rafe pf 4il.per Cent.
Compound Interpft ?
Here P=275,75 ; A=32i,68; R=?i.p45; tp find
t, per Theor. IV.
. From the l^oe^ithm of . , A=:3?i,68=:2.50742i
Subduft the Logarithm of 1*=:275,75=:2.4405I5
J'^e P'lff. is the Log. of ... . Rt=i,p4^4=:p.66996
XbentheLpg. pf 1.045=50.01511 6)0.066906(3, 5=:t.
'..;..:' . 57348
'— 955S0
95580
Thus the Anfweris 3 Years and 6 Months.
iV#/tfi As the Amount of any Sum andf its Intereft is
greater at Cemppund Intereft than at Simple Int^eft
for any fitne above a Year j fo it is /^ at Com-
J
Cbap. VI. ^;^ Logarithms. j^^
pound than Simple Intereft for any 5iW Ufs than a
Year, a5 the Learner may eafily ptovtf by tile
Theorems beforegoing.
5. Cy Annuities, £s?r. in Arrears.
Theor. I. r~— = A = the Amount of any Sum.
Thtor. 11. ^Cli = U = the Annuity^ Penfion^
&c. ,
Theor. III. ^^^^^=^ = Rt == the r/)^.
; Thcor.IV. ^ «g-,R — R^ = thc Amount
ofi/.
6. Queft. L If 30/. yearly Rent be forborn 9 Years,
what win It amount to at 5 /. per CenL Scc^
G>mpoqnd Intereft ?
Here U=^o, t=9, R:5=i ^05 •, to find A ; per
Theor. 1.
The fcogapthin of , . . . R=i,05=o.o2 1 1^^
multiply by tlic Index ... g
The Produft is . . . '. '. *. [ R*=i.5i6=ro. 1^0761
To which add tfie'Ldg: of. . . 11=30=1 .477 1 2 1
The Sum isthc Log. of UR'=45,48=i.657822
Subduft.... U=36
* I '
There reinainis . . . UR*— rU=i 5,48=1 . 1 8977 1
Subftra«a the Log. of K— 1=0.05=1.8. 6g8gyo
Tbcpiif.isthej UR'-rU . ^ ^ „
Log,of rT^ = A = 309,6=4.490801
The Amount therefore is 309 /. 12;. foj- Anfwer.
Que(t.
142 Compmmd Inter eji
Qjiqft. n, Whjtt Annqity, iSc. wfll ftiifc a Slock (or
•mount to) 309/. lis. being forbom or un-
paid 9 Years, at 5 /. ftr Cent. &c. Compound
Intereft?
Here A=:309,6 •, t=^ ; R=i,05 } to find U, per
Theof. II.
A^ S the Logarithm of ... . A=309,&=:2 .490801
^ I the Logarithiit of . . . Rs=i ,05=0.02 1 1 89
The Sum is the Log. of . . AR=s^5,q8^2.5i 1990
Therefore the Log. of AR — A==i5,48=:i.i8977i
Subduift the Log. of". . . R'^—i=xa.5 162=19.7 12650
TheDifF.isthej AR->A ^^ ...^771177
Lofrof f • • • TKCT =U = 30==i.477«i
Th^t.is. 3pA /ff .^94p^i& the Jnnuity ipu^t.
Queft. HI.. In what time will 30/. fer Annum a-
•mount to '30^/1 T2J. ztgh per Cent. Com-
pound Intfereft ?
Here U— 3P ; Ai£309,6 1 Rdtr 1,05 5f to^d t, ^fr
Theor.m. • •• / V
^tEe Logarithm of . . . • A=309,6==2 .49.0801
*Tthe Logaifthm of] ^; ; .'. Rz=i,o52=6.02ii89
The Sum i& the Lpgr of AR=?3Z5,98=;? .511 990
Subftraift .... ^ . A— U=279,6
There remahir "^R.^U.^A=45,48=i.657822
Sufcttraft the Ldgr of - * • U= 30=1.47712 1
The^DHF. is the Log. of . . R'=:ivQ5^=o.i8o7oi
Then the l^og^of i,(at5Q.02;;5i i$9>0ii8o7Oi(9=st.
■ "' ^ I . 180701
•I > ■
AriC\Ycr, 9 Yeys. .....
. ^. " Queft*
Chap.VL 4y '-'^^^^^'^"Ms. m
Qucft. IV. At what Rate per Cent. Compoimd In-
tercft will 30/. Yearly Rent amount to
309A i2s. in 9 Years?
Here U=30 j A=309,6 ; t±=:9 ; to find R, pet
ThconlV.
From the Log. of . . . A— U:=:2 79,6=52446537
Subduft the Logarithm of . • . 11=303:1.477121
A— U
The DifF. is the Log. of --^p =9,32=0.96941^
Agsun from the Log. of . . . A=309,6=2.49o8oi
3ubftraft the Lc^^thm <^. . 17=30=1.477121
Thel^. Isf he Log. of ^ =10,32=1.013680
Therefore the faid Theorem IV. is ftduced to this
Equation, viz. 9.32=io,32R — R\ Now this E-
quation may be eafily refolv'd by a converging Series j
or ftill eafier by the Tables of the Amount of i L
for femoral Years fucceffivcly ; for therein R% or the
Amount of i /. in 9 Years may be tried for the fevc-
nd Rates p^r Cent, in the Tables. For inftancc, fup-
pofe I pitch upon the Table at 5 1, per Cent. Com-
pound Intereft, then I find R'=i.55i328, fcfr.
Wherefore 9.324'i*55^3*8=io.87i328=io.32R ;
therefore 10,32) 10.871328 (=1,05. Now fince
i7. : 0,05/. :: 100/. : 5/. the very Ratep^r Cent, af-
fumcd in the Tables, it will follow, that is the true
Rate fought.
Note^ This Qgcftion may be refolved by theRul^
of falfe Pofition 5 which I leave to the Leamcr^s
Ewrcife.
144' Compound Intefejl
»
7. Of the Present Worth of Annuities,
tf r, at Compound Inter^.
Here P denotes the prefent Wortf} of any Annuity,
Leafe, 6f r. and the reft of the Letters as before.
Art. 6. IThen the Theorems are as follows.
Thcor. I. ..J— iL =: P = the prefent Worth.
Theor. II. — ^^^ = U = the Annuity^ &c.
Thcpn III. t>4.ji,^R = R' = the ^ime.
Thcor. IV. ^ =^ R^ + R'—RS-?=: the Value
ofR.
Queft, \ . What is 30 /• per Annum to continue 9 Years,
worth in ready Money, abating 5/. per Cent.
&c. Compound Intereft to the Purchafer ?
HcreU=30, t=9 j R=i,05 5 to findP, per
Theor. I.
The Logarithm of ... . R=:ij05=o.02 1 189
Multiply by the Time .... 9
The Prod, is the Log. of R5'=i.5i6=o. 180701
which fubftraA from the Log. of U=30=r.477i2i
»* ■ ' " ■ ■ , ■
The DifF. is the Log. of j*^ =i9,'788=:i.29642ro
which take from ... 11=30
there remains ... U — j;;^ =10.2 1 2 =1 .0091 1 1
Subdudthe Log. of . . . R— i=o.05=.8.698970
• The DifF. is the Log. of P=:204,24=2.3ioi4i
The prefent Worth is 204/. 4^. 9 J. I, theAnfwer.
Queft.
I
Chap; VI- hy LbGARitftMS. 145
Queft. li. What Annuity, t£c. to continue 9 Years
may one purchafe for 204/. 4/. gJi. abating
5/. fer Cent. &c. to the Purchafcr ?
Here P=204,24 ; t~^ \ R=i,05 ; to find \5if&
Theor. II.
r the Logarithm of .. ; . R^r= 1.5 i6z=:0. 180761
Add ^ the LogaHthm of . . . . R=i,05=:O.C2ii89
tthc Logarithm of . • P=2d4,24£ss2.3ioi4i
The Sum i^ the Log. of PRR*:it:325, 11=2.512031
Then from . . . • PRR*=325i,ii
Subftrad ; : . . ; PR*=i:309.628
there remains PRR'—PR*=i5,482=i. 189799
Subduathe Log. of R'— i=:o.5i6 = .9.712656
The Diff. is the Log. of ; . . Ue=30==i.477i49
Therefore the Annuity fought is %oLprAnnumi.
Queft. III. What tirhe may one enjoy an Annuity^
fcfr; ef ^ohper Annum;^ for 204/. 4 J. ^dh
ready Money, abating to the Purchafcr 5/^
per Cent. dccJ
Here P:^204,24 ; 17=^30 i R=j:i.d5 i t^ ^^ ^^
per Theor III.
aaaS ^^^ Logarithm of . . Pfe2d4,24=2,3ioi4i
-^^^ithe Logarithm of R=ii,05i=d.02ii89
the Sum is the Log. of Pk=i=2 14,45^^2.33 1336
Then... .i P±U=i234.24
The DlfF. . . . P4.lJ^PR=£:ti9.788===i.2964t)2
ifrhich fdbftl^a from the Log; of U==3d=i.477i2i
thcDiff.tstheLog,6fRH=:l,o5^=i,5i6=o.i8o7i9
U Then
146 , Compound Inter efl
Then the Log. of 1,05=0.02 n 89)0. i8o7i9(=9=it.
I 8070 I ^
Anfwcr, 9 Years. 18
Queft. IV. Suppofe an Annuity, t^c. t>f 30/. per
Annum^ to continue 9 Years, be fold for
204/. 4.5. gdi. yready Money, what rate of
Intereft hath the Purchafcr allowed per Cent.
&c. for his Money ?*
Here P=204,24 ; U=3o •, t~9 ; to find R, per
Theor.IV.
From the Log. of the Annuity U:==: 30^1 .477 12 1
Subduft the Log. of . . . . P— 204,24=22.310141
-•■ •
The Diff. is the Log. of -p =o.i4579=:.9. 166980
Then the faid Theor.lV. is reduced to this Equa-
tion, 'y/z. 0.14679=^0. i4679R5-f.R^ — R'^5 which
is to b^ folv'd by the Method of Infinite Series. But for
thofe whounderftand not that Method, this Queftion.
h much better anfwer*d by the Rule of Falfe j or yet
eafier by the Tables of Compound Intereft. See a
compleat Set, with their Ufes, in my Syjiem ofDeci-
fj$al Jrithmetic^ publiQied for Mr. Noon^ at the IVbiie
Hart^ mCbeapfidey London.
8. OfPurcbaJing Freehold Estates.
FJtates in' Fee Simple (yjhkh are fuch as we com-^
monly call Freehold or Real Eftates) being pur*
chafed Jor ^ver^ or without Reverjion \ 'tis plain,
that in the foregoing Theorems for finding the pre*
fent Worth of Annuities, (^€. if thofe Terms
wherein t (= the Time) is found, be made to va-
niih» aa being Infinite ; the faid Theorems will be
reduced
Chap. VI. hy LoGARiTftMS.^ f47
reduced to fuch as fuit tht prefent Cafe^ and arc
as follows.
Theor. I. ^^ = P, the PrefintfFortb^ or Purchafe^
Money.
Theor. 11. PR — P=U, the Armuiiyy or Efiate per
Annum.
^hcor. III. .^ =: R, the Amount of i/. at the
given Rate.
Quefiion. What muft be given for a Freehold Eftatc
of soLper Annum^ allowing the ^uycr 61.
per Cent. Compound Intereft for his Money ?
fiere U^soy R=i,o6 5 to find P, p^r Theor. 1. *
The Logarithm of . . . . 11-50=1.698970
Subduft the Log, of R4-i=rO,G6=.8.778i5i
The piiE is theLog. of ^ =P=833,j-=2,92o8 19
< The Purchafe- Money, then, is 833/. 6s. id. If
\t had been atthie. Race q( sLper Cent. Compmtnd
In^r^Y the faicl Eftate would be worth lOOoA pre-
fent Money i which is equal to' twenty times the
Yearly Renf: ^hd therefore the* general Method of
buying Freehold Eftates^ v& by paying 20 or 25 times
the Yearly Rent ; which is commonly called 2q or
25 Years Purchafe.
The other two Theorems are wrought in the fame
Manner for U and R j which, being fo very eafy,
«ieed no ^.xa^iple.
U 2 CHAP.
148 Vulgar FMStiQm
CHAP. VII.
Cy Vulgar Fractions ^jr Logarithms*
i.fTp
O find the LogarUbm of a Vulvar Fraffion^
this is the
iFrom the LDgarithm of tht J^TumerAtai^
Subftraft the Logarithm of the Denominator 5
the Remainder is the Log. of the Fraftion. .^
Exampl? I. What i$ the Log. of the FradUon ^3 ?
From the Logarithm of . . . 5^^^-707570
Subftiad the Logarithm of . • . 73=1,863323
there remains the Log. of H'=o.69862==.9.844247
Exam. II. What is the Logarithna of 77^5-6 ?
From the Logarithm of ... . 13=0.000000
Subftra£t the Logarithm of . • . 1 7562=3.244524
there remains the I , _ rsrs^r^r,.^— #c *,^^^^<
Log. of r • "^V^ - ><^o^56947=T.P/755i76
Exam. III. What is the Logarithm of r^, >
From the Logarithm of ... . 100=2.000000
Take the Logarithm of . . , 1 357= 3 . 1 32589
there remains the Log of-^==,073692=.fe.867420
Exam. IV. What is the Logarithm of r^
eo '
?
T¥
C^p.VII. ^y Logarithms.' 149
The Logarithm of ; . . . 5973=3776192
From which take the Log. of ioooooo=6.poopoo
there rem. the Log- ofT^^=,oo5g73=.7.776i9^
2. If the Fraftion be a mi^done^ it muft be re-
duced to an improper FraSion \ apd then procce4
wit|i it as before.
Example L What is the Logarithm of 13I ?
fiere I3f=f 5
Therefore from the Logarithm of 96=1.982271
take the Logarithm of . . • . 7=0.845098
there remains the Log. of 131=13.714=1.137173
Exam. II. What is the Logarithm of 193^ ?
Here 193,4=1^;
Then from the Logarithm of 14140=4,150449
take the Logarithm of .... . 73=1.863323
fhererem.tl^c Log, of I937i=i93>^986?=2.287i26
3. If the m\x'd Fraiiion confifts of large Num^
hefSj it may be moft eafily reducM by Logarithms,
thus 5 fuppofing the Example be 2145^*-^ j
To thp Log. of th€/«/^jfra/Piir/ 2145=3.331427
iVdd the Log[. of the Dehominator 589=2.7701 15
The Sum is the Log- of . , 12^3405=6.10154:2 .
to which add the Numerator . . 57
ithe new Num. 1 263462 7 .. t.^^
theDenomin. S7 3^^"SW.
* . its *' •
4<
t^o Fui^ar FraStions
4. Ta mtdfipfy Vulgiir Fraffions hj Logarithtnsi^
tdd the Logarithms of the Numerators for the Lo-
garithm 0f a new Numerator ; and the Logarithms of
the Denominators for the Logarithm of a new De-
nominator*
Example. What is the Prbduftof -j^^ into -J ?
AddtheLog«.oftheNumerators|' 35=;;|440^^
The Log. of the new Nuinerai»r 245=^2.389166
Add the Log., of the Denominat'. j '^P^'^f^^t^
The Log. of the ne'iv l)enoinin. i557=;=:3.i9228S
Therefore ^ x J =c: ifti^ the fradional Produft
required.
5. The Logarithm oftlfeProdu^ offevera! Frac-
tions muitipiied into one another is thus obtained ^
c/2.- Add the Logarithms of all the Numerators and
the ^tbmetwai Complements of the Logarithms of
all the Denominators together ; the Sum is the Loga-
rithm required.
^M^ What is the Logarithm of ^-^ :?c H x t ?
-; • 3 the ]^pgirithms of , < 33=1.518514,
Addto-J^ 'I 3===o.477i2^
l^^^-.WAnih Co.p .^the ^^:^
Chap.Vit 1^;^ Logarithms, 151
€. To divide Vuigar Fradiohs by Logarithms^
do thus; Add tlic Logarithm oF the Denominator of
the Divifor to the Logarithm of the Numeratvr of
. the Dividend \ the Sum is the Logarithm of a iHVf
Numerator \ and the Sum of the Logarithms of the
Other two FaSors^ is the Logarithm of the new De-
nominator of the Quoticat required.
Example! Divide ^ by h
AddtheLogaritkmsofj .^'S^f^^S
Thfe Log. of the new Numerator 2205=3.343408
Add th. Logarithm, off . . . ,J^ g;^^g||
The Log, of the new Denom. 10899=4.037386
therefore i).^,(«^^,=^. See Art. 4.
7. The Logarithm of this Quotient tnay be found
by one Addition, in like manner as dittfted in Art. 5^
thus ;
gether j the Arith.Comp. ofc 7=.9. 154902
t the Logarithms of I i557=.6.8o77ia
The Log. of . , i)rrf;(=:^,=,2023i=:.9.3o6o22
B
f 8. The Extraaion of the 2?i9(?/j of VuJf^ar Frac-
I » tions by Logarithms is thus performed. Divide the
Logarithm both of the Numerator and Denomihatot
of the given Fraftioti by the Inde:^ of the Root ; the
^otients Ihall be the Logarithms of the Numerator
and Denominator of the FraSional Root required.'
I . Examp.
152 Fiitgar FradHont
£xamp. I. What is the Square Root of the Fraftiofi
SOlOl *
Divide the Logarithm of ..... 1849=3.266937
by the Index of the Root ... 2
The Log. is the Num. of the Root 43=1.633468
Again, divide the Log. of. . . 10261=4.008643
by . 5 i . ; . 2
ThfeLog. of the Uenom. of the Root 101=2.00432 1
Therefore v"^^ sc 1^ the Rodt required.
Example IL What is the Cube Root of 86*-|i .?
This reduced to an impropcf Fradliorii Is ^f^' j i
Therefore the Logarithm of 29791=4.474085
i thereof is the Logi of the new Num. 3 1= i .49 1 3 6 2 \
Again, the Logarithm of . . .". 343=2.535294
T of which is the lx)g. of the »^wZ)^»^w. 7=0.845098
Confequently ^86j|f = 't = 4^, the Cube Root i
fought*
^ 9. To find the Logarithm of the Root of any Frac- \
tion ; add the Logarithm of the Numerator to the
Jritbmetical Complemehi of the Logarithm of fhe
Denominator^ and divide that Sum by the Index of
the Root \ the Quotient fhall be the Logaritfim
fought. ^ .
Example L What is the Logarithm of the Square
Root of theFraftion -^'i ?
the
f
>
Chap. Vil. iy Logarithms^ i^j
The tx)g. of the Numerator . . • 1849=3.266937
Gomp. Arith. of the Log. \ ,^^^__ ^
of the Denom. } • • I020«=.5.99i357
The Sum is .9.258284
which divide by the Index of the Root 2
The Quot; is I y-trrs- *^ ^r
theL:og.of } /t.^.=*1V.«42574=.9.629I47
Exam, IF; What is the Lb^rilhih of the Cube Rooi
of the mix'd Fhuilion 863*11, or its equal
»079i i
A jj Cthe LogaPithhlof*. . ; ; 29791=4,474685
\ the Ar. Com^ of the Log. of 343=7.464706
The Sum is the Log. of .... . 681^1=1.93879*
^hich divide by the Index of the Root . • 3
c ii A P. viiL
Duodecimal k^iT'^w^tiQ perfirmed by
LbOAfelTHMS.
i. Qj I N CE this kind of Arithmetic is fo very co)n-
\j mon^ and yet in the common Way fd very dif-
ficult ; I hdpe 'twill not be unacceptable to the yeung
Artificer to be convinced with bow much more Eile
land Pleafure he may compute his Dimcnfions in this
Way by the help of Logarithms. And as it is pro-
p&T to reduce them firft to Decimals^ I have here fub-
. . X joined
'54
Duodecimal Arithmetic
joined a Table (hewing by InfpeAion the Decimal
Parts of a Foot (in this Cafe the Integer) anfwering
to any Number of Primes^ Seconds^ and Thirds -^
which are the Parts or Diviiions of a Foot made ufe
of in this kind of Menfuration.
-^2.
The Duodecitnal Table.
Duode-
cimals
I
2
3
4
5
6
7
8
9
10
II
Decimal Parts.
Primes ' ISecondsl Thirds
»o8j333
^1^6666
J33333
,41^666
5
>583'333
,^66666
75
^^213333
,916666
,006944
013^88
,02o83'3
,02^777
.0347A
,041^66
,048611
og5555
,0625
,069444
n///
000578
,001157
,001736
,002314
002893
,003472
,004051
,004629
^005208
005787
,006365
3. The foregoing Table is too eafy to need De*
icription, I mean for any Perfon concerned in Duo-^
decimal Menfuration ; and therefore I ihall proceed
to exemplify and illuftrate the Operations this Way
by Logarithms in all the Rules of Multiplication^
Divifiotij Involution^ and Extraction of Roots.
4. Mutiplication ^2/* Duodecimals by Loga-
rithms.
Examp. I. Suppofe a Plane be 9^ 10' in Length, and
8^ 8' in Breadth v ^^re the Content or Area ?
^, , c the Log. . . of 9^ io\=9. 83^3 3=0.99 2 704
•^^^itheLog.of.. 8f. 8'.=8.6666=o.937849.
The Sum is the Log. of . • . . 85,;2f22asi.93055g:
Thatis, 85,222 Feet,=:85^ 2'. 8^ the Area requirecL
Examp
Chap.Vli. ^Logarithms. 155
Examp. II. What is the Produft of 40'. 9'. 10* by
ii'.9'?
- , , c the L. of 40f. 9'. io'.s=40.8 194= 1.6 10866
-^^"l the Log. of 11'. 9'=o,979i6=!.9.99o854
The Sam is the Log. of ... . 39,969=1.601720
Therefore 39,969^39^. 11'. y'.S'". the ProduS,
or Area required.
Examp. in. What is r 75 Feet 00'. 04'. by 8'"?
. , . 5 the L. of 175'. o'. 4*.=i75,o2^=2.243io7
^^** I the Log. of 8"'.=,oo4629=. 7.665487 .
The Sran is the Log. of . . ,8/1 1 1 =.9.908594
Therefore o,8InI=o^ 9'. 8*. 9"' = the -rfr^«
fought.
Examp. IV. What is 17'. 9'. 2". &" by 6'?
.,,jtheL.ofi/.9'.a*.6'"=i7.76736=i.2474i.7
^'***i the Log. of .... 6 ^0778151
The Sum is the Log. of . . 106,60416=2.025568
Thusio6.6o4i6=io6f. 7' 3'. the Area fought.
Examp. V. What Number of 7&/f^ Feet is in a Ctllar
2if. 2'. long, I If. 10'. 8'. broad, aad 7f. 3'.
deep?
r the Log. of. • »»'• 2'=2i>i06=i.325659
AddStheLog.of iif. 10'. 8'=ii.?88=i.o74io9
ttheLog.of 7'- S'- =7»g5' =0-«^0 338
The Sum is the Log. of . . . 1820,19=3.260106
Therefore i820,i9=i820f. 2'. 3'. 4'", the SelidHy
required.
X 2 . 5-
1 56 Duodecimal jirithmetic
5. Division ^/Duodecimals^ Logarithm^.
pxamp. I. What is 85^. 2'. 8^ divide by 8^. 8' ?
The Log. pf • . . . Sgf. 2{. 8^=:85,;2f2=i. 930553
SubduA the Log. of 8^. 8' =8,i|?66=o.937849
rr
The DiflF. is the Log. of . . . 9 -83^3 3=0.992 704
Sothat9.83'33=9^. 10', theAufwer.
Examp. 11. What is 9'. 8^ 9^ divided by 8''^?
From the Log. of 9^ 8^ 9"'=p.8xiii=.9.9p8594
Subftraft the Log, of 8'"=,oo4629=.7.665487
The Diff. is the I^og. of . . .. 175,02^=2.243107
Therefore the Anfwcr is 1 75,02;/=! 75f. o'. 4.".
Examp. in. Divide 39^ 1 1'. 7^ &". by 40*: 9'. lo^ '
^Loltf }• • 39^- XI'. 7^. 6^''=39.9687=i.6oi720
Subd. the Log. of 40^. 9'. 10^=40.8194=1.610866
The DifF. is the Log. . . of pr979i6=.9.99p854
But ,97916=1 1'. 9^. the Quotient required. ]
Examp. IV. pivide 106^. 7'. 3^. by 6,.
From the L. of io6f 7'. 3^,=io6,6o4i^2. 205568
Subftraft the Log. of 6=0.8 78 1 5 1
* The DiiF. is the Log. of 17,76736=1 .247^17
Jhus 1 7,76736=1 7^. 9'. 2". 6"'* the Quotient
fought.
^.
chap. VIII. /^Logarithms. 157
0. I NVOLuTiON pf Duodecimals iy Loga*
RITHMS.
Examp. I. What is the Area of that Square whofe
Side is 12^. 9'. 7^ iq'^'?
TheLog.of 12'. 9' 7^. xo'^=i2.8p439=s:i,io7359
Multiply by . • • 2
The Produft bthe Lc^. of 163.9524=2.214718
Therefore 1 63.9524=1 63^. u^ 5^ i''>. the ^r^tf
required.
Examp. II. What is the Solidity of a Cuh whofe
Sideisi^2'. 9^ ii''^?
The Log. of if. 2'. 9^. I i''^=i,23553=;o.09i854
which multiply by , . . ^
The Produft is the Log. of 1,877/1=30,273562
Therefore f,8774=i<'. ip^ ^^. 4^ the Solidity
fought.
7. Extraction (J^Roots <?/ Duodecimals
^Logarithms.
Examp. I. What is the Side of that Square whofe
Areai8i63f. 11'. gf. i^^i
The L.of 163. 11'. sf. i'''.=i63.9524=2.2i47i8
Divide by . . . 2
The Quotient is the Log. of 12.804392=1.107359
Thus 12,80439=1^^- 9'- 7^- 10'^'. the Side fought.
Examp. II. What is the SiJe of that Cuk, whofe
Solidify is if. i&.6^.4"'i
The
t^S Commm Ruksofjiriibmetie
The Log. of X. lo'. 6^. 4'".=i.S774=20.a735(Jft
Divide by ... . 3
The Quotient is the Log. of I.23553=:0.09i854
. But 1,23553=1^. 2'. /. ii'^ the Side of the Cube
required.
Thefe few Examples abundantly (hew with how
much more Eafe^ Brevity and Expedition the Opera*
tions of Duodecimals are performed by Logarithms^
than by the ordinary Method.
CHAP. IX.
Ti&^ OpfeRATiON of the common Rules ^A-
RiTHMETic ^j^ Instruments; viz. the
Logarithmic Scale ; and Gunter'^
Line, noitb the Compasses, and on the
Sliding-Rule.
I. TTAving in the fbeory /hewn the Nature and
Jl X ConftruBion of the Logariibmic Scakt and
hunter's Line 5 I (hall here briefly cxAplify their
Ufes in the Operation of the common Kuks of A-
rithmetic thereby; and in doing of this I (hall ob-
ferve this Method ; firft, to give the Operation by
Logarithms in Numbers. Secondly,^ to perform the
fame by the Logarithmic Scale. Thirdly, to work
the fame Cafe on the Gunter with the Compajfes ; and
fourthly, to do the fame thing on tht Sliding-^Gunter.
In this Method, the Analogy or ^reement between
the Numerical ahd Iifftrwnentat Operations will more
eafily appear; and the Nature and Reafon of the
latter be much better underftood hy young Learners.
2^
Chap. IX. hy iNSTfttJUXNTs. 159
a. Multiplication^.
Examp.!. Multiply 9^7*
Firft, by Logarithms.
^ J , 5 the Log^thni of 9=^.954241
^^^ I the Logarithm of 7=0.845098
The Sum is the Log. of the Prod. =63=1.799340
3. Secondly, by the Logarithmic Scale.
(Ifote. If the fmalkft Divrfions in the Line AD
reprcfcnt Numbers^ the Logarithms begin from C c
in the Line Ae } if the middle Divifions in A B be
Numbers, the Logarithms begin from G 5 but if the
hrgcft Divifions in AB be taken for Numbers, then
the Logarithms begin from H, in the faid Line A e^
And fince xkz finaUejUhvifions are toofmall^ and the
largefi Divifions too large for Examples, we muft
coniequently chufe the mean Divifions in A B to rc-«
prefent the Numbers 1, 2, 3, 4, (^e. or 10, 20, 30,
40, 6fr. or lOo, 200, 300, &r. and fo the Loga^
rithms begin from G. Therefore) Set one Foot of
the Compailes in G, and extend the other to the
Logarithm of the Multiplier 7=ag, which you^l]
find to be Ga=845, and fince G b=954 is the Lo^
garithm c£the Multiplicand 9=bh ; therefore with
the fame Extent G a in the Compares, fet one Foot
in b, the other will fall on d ; therefore Gd is the
Logarithm of theProdud: dm=63 xn A B, the Num^
ber iought.
4. Thirdly, by the Gunter with CompaflTes.
Set one Foot of the Compafles in the Beginning of
«iie Lmeat i, and extend the other to 7 ; with that
Extent in the Compafles fet one Foot in 9, the other
witt fikU Oft -^3, the Produ<3: fequiired^
NoiK
i6o Commoh RuksofArithfHetid
Note. When the Numbers zxtfmall the larger l)i-^
vifions may be ufed, as in the prefent Cafe ; but il
the Numbers be large^ the lejfer Divifions muft be
ufed.
$. Fourthly^ by the Sliding-Guntef.
In this Cafe, there is one Line of Numbers on the
Rute^ and another on the Slider^ both mark'd with
N, at the End. And it is eafy to conceive that by
Aiding one of thefe by the other, the fame Effefts are
poduced as before with the Compares ; that is, any
Part of the Line on the Rule is transfer^d to, or com-
pared with any other Part of the faid Line by means
of the Aiding Line.
Therefore fot i on the Slider to 7' in the Line on
the Rule ; then againft 9 on the Slider, you find 63
on the Rule, and that is the Produdb fougkt
6. Examp.II. BytheG«/i/en WhatistheProdu^E
ot27 by 18?
Here the leAer Divifions mtift be ufed, aiid the
greater ones reckoned 10, 20, 30, 6ff . ort thtfirjl
Radius ; and confcquently on the fecoHd Raiiiis they
will be 100, 200, 30o,.£s?f. if the double Radius be
ufed. For then it will be 10 : 180 :: 27 : the Produdt
fought. But fince 10 : ,180 :: i : 18 ; therefore if you
make i : 18 :: 27 : the Produft 5 the Jingle Raaiui
will give the Anfwer in the fame manner ; only re-
membring that th^ fourth Number fought will be of
the fame Denomination with the fecond^ which in this
Cafe is Hundreds.
Therefore fet one Foot of the Com{)a{les in the Be-
ginning of the Line, and extend the other to 18, the
ume Extent will reach from 27 to 486^ the Produ6fc
fought.
And by thtSliding'Rule^ thus; fet ion the Slider
to 18 on the Rule, and then againft 27 on the Slider
you
I
Chip. IX. 4y Instruments. . i^f
you find 486 on the Rule, which is the fame Producft
I as before.
^ 7. Examp. III. What is the Produdt of 257 by 34 ?
Take in your Compafles theDiftancc from i to 34
on the Line of Numbers^ the fame Extent will reach
from 257 to 8738-, thie Produft required.
By the Sliding- Rule, thusi Set i on the Slider t^
34 on the Rule, and againil 257 on the Slider, yoa
fee 8738 on the Rule^ which is the Produft as
before.
S. Examp. IV. What is the Produd; of 215 by
108?
With the Compafles, take the Diftance from i to
ko8 on the 'Gunter^ the fame Extent of the G)m-
pafles will reach from 215 to 23220, the Produft re*
quired.
By the Sliding Rule, thus \ Set i on the Slider to
7 108 on the Rule, andagainft 215 on the Slider you
find 23220 on the Rule, the Produdl fought.
9. When the Produft becomes fb large, it muft
be a v^ry large Line of Numbers indeed to (hew it
near the Truth ; the Ufe of thefe Lines being prin-
cipally where the Numbers arefmall-, or where great
Exaftncfs is not required. They who underftand
the foregoing Doftrine of Logarithms can never be
at any lofs to know how many Places of Figures are
contained in the Number fought, in this, or any of
the following Rules.
to. Division.
Examp. I, What is the Quotient of 63 divided by 9 ?
Firft, by Logarithms,
y ' From
1 62 CofHmon Rules of Arithmetic
From the Logartthm of . . . 63=1.799340
Subdud the Logarithm of . . . ' 9=0.954242
The DiE is the Log. of the Quot. =7=0,845098
tu Secondly^y by the Logarithmic Scale.
From Gd=i799 the Logarithm of dma=63, take
Gb=954 the Lc^arithm of bh=9 ; and there will
temain Ga=845) the Logarithm of ag=7, the
Quotient fought
12. Thirdly, by Gunt&^s Line and CompafTes*
Set one Foot of the Compafles in i, and extend
the other to 9, and then with that Extent of the
Compafles fet one Foot in 6^^ the other will fall (to-
wards the beginning of the Line) on 7, the Quotient
fought.
13. Fourthly, by the Sliding-Rule.
Becau(e 9 : 63 :: i : the Quotient ; therefore fet 9
on the Slider to 63 on the Rule, and then againft i
on the Slider is 7 on the Rule, which is the Quotient
fought
14- Examp. II. What is 486 divided by 18 ?
By the Gtinter and Compafles.
Extend the Compafles from i to 18, that Extent
will reach from 486 (downward) to 2 7, the Quotient
required.
By the Sliding-Rule.
Set 18 on the Slider to 486 on the Rule, then a->
gainft I on the Slider you find 27 on the Rule, the
Quotient fought.
All other Operations of Diviflon being performed
in the very fame manner, 'tis needlefs to add any
more Examples of this kind»
.'5-
Chap. IX. hy iNSTRUMENts. \h\
15. iNVOtUTJON.
Examp.!. What is the Square of 9 ?
Firft, by Logarithms.
The Logarithm of . . . 9=0.954242
Multiply by the Index . . • %
The Prod, is the Log^ of the Square =8 1=1 ,908484
iS. Secondly, by the Logarithmic Scale.
Let the Logarithms begin from G in the Line
A e, as before 5 then with the Compafles take thp
Diftance gb=954 the Logarithm ofbhi=PB9; and
with one Foot remaining in b, turn the Compafles,
the other Foot will fall on n; then fliall Gn=a:
(aGb»)i9o8 the Logarithm of n 0=81, whiclj
therefore is the fquare Number fought
17. Thirdly, by the G«»/^r and Compafles.
Set one Foot in i, and extend the other to 9, wherq^
keep it fix'd, and turn the Compafles, the other
• . Foot will fall on 8 1, the Square fought.
18. Fourthly, by the Sliding-Rule.
I Becaufe i : 9 :; 9 : the Square required, therefore
fpt I on the Slider to 9 on the Rule, then againft .9
' on the Slider is 8 1 on the Rule ; whi<?h is the/^uare
Number (ought.
19. Examp. II. What is the Cube of 9 ?
t . By the Gunter and Compafles.
'- Extend the Compafles from i to 9, that Extent
will reach from 9 to 81, a^d agai^ from^ 8j to 729,
the Cube Nunxber required.
• Ya Br
i
1 64 Common Rula of Arithmetic
. ■' .
By the Sliding- Rule.
Set I on the Slider to 9 on the Rule, thep againft
9 on the Slider is 81 on the Rule,' and againft 81 on
the Slider (remaining unmov'd) is 729 on the. Rule,
the Cube Number required.
20. Examp. III. What is the Square and Cube of
the Number 37 ?
By the Gunter and Compafles.
Extend the Compafles from i to 37, that Extent
•will reach from 37 to i '^6^ the Square \ and the fame
Extent will reach from 1369 to 50653 the C«^^ 5
both a3 required.
By the Sliding-Rule.
Set I pn the Slider to 37 on the Rule, then againft
37 on the Slider is 1369 on the Rule, which is
Square \ and againft 1369 on the Slider (remaining
unmov*d) is S^^Si ^^ ^^e Rule, which is Cub^ of
37 5 both as before.
21. Extraction of Roots*
Examp. I. What is the Square Root of 81 ?
Firft, by Logarithms.
The Logarithm of ... . ' 81=1.908484
which divide by the Index • . . . z
The Quotient is the Log. 7 ^ ^ «^.r. .^
ofti:.SquareRoot ^t ....=9=0.55424^
22. Secondly, by the Logarithmic Scale.
Bifed Gn=i9o8 the Logarithm of no=8i, in b ;
then (hall Gb=954 be the Logarithm of the Square
Root, viz. bh=9, the Number fought.
«3-
chap. IX. 4y Instruments. -165
23. . Thirdly, by the G«;«/^r and Compafles;
Take with the Compafle? the Diftance between i
and 81, and bifeft it; then take one Half in the
Compafles, and it will reach from i to 9, th^fquare
Root fought.
24. Fourthly, by the Sliding-Rulc.
Move the Slider forwards and backwards till you
make the fame Number on the Rule anfwer i on the
Slider, as on the Slider anfwers 8i on the Rule;
which Number will be the fquare Root fought, and
in the prefcnt Cafe will be found 9.
25. Examp.^II. What is the Cube Root of 50653 ?
Divide the Diftance between i and 5065^ into 3
equal Parts ; the firft Divifion will fall on 27 j the
Cuie Root required.
- Noie^ In the double Line of Numbers, if the
grand Divifions be efteem'dt7»i/j in the firft Radius,
thofe in the fecond Radius will be Tens ; if thofe in
the firft be Tens^ viz. 10, 20, 30, (^c. thofe of the
fecond will be Thoufands^ as 1000, 2000, 3000, (^c.
with regard to fquare Numbers : and confequently in
- ExtraSion •, if the Number whofe fquare Root is
fought be lefe than 100, yet greater than 10, the
Number it felf will be found on tliQ fecond Radius i
and its Root a Number of Units onthtfrft Radius.
But if the Square be lefs than 10, both it felf and
Root will be found in tht firjt Radius.' Again, if
iht fquare l^umber h^'htVNG^n 1000 and loooo, the
Number it felf will be found on the fecond Radiits^
and its Root a Number of Tens on the firft Radius^
But if it be between 100 and 1000, both the Num-
ber and its Root of Tens will be found on th^firfi
Radius. After the fame manner you may rcafon
concirnbg the Cube Number and its Root.
i66 Common 'Rules of Arithmetic
26. Since the Logarithm of the Square h dduhl$
the Logarithm of the Root ; and the Logarithm of
the Cube triple the Logarithm of the Root : therefore
if a Line of Numbers of a ^ngle Radius, be equal to
another of a double Radius^ and thefe two appofitely
laid together, beginning from the fame Point ; thea
againft any Number on th& Jingle Radius^ you fee its
correfpondent Square on tht double Radius i andfuch
Lines you have on fome Sliding- Rules.
Alfo if a Line of Jingle Radius^ were made equal to
another of a triple Radius, and thefe exa^ly and
properly placed together, then the Numbers on the
latter would be the Cubes of thofe on the foimer;
and fo the Square and Cu^e Roots of ^ny Number j
and vice verfd, would be difcoverable by Jn/peff ion.
%"]. Moreover by means of ^ Jingle and double Line
of 'Numbers made to Aide by each other, *tis very
cafy to find a mean Proportional between any two
given Numbers ; as fuppofe 13 and 23. .Thus; fet
, 13 on the double Line to 13 on the Jingle one^ then
againft 23 on the double Line is 17,35 on the Jingle
cne, which is the Mean required between 13 and 23.
Or if you fet 23 to 23, then againft 13 on the double
you find 17,35 on t\iz Jingle Line, the A&<?» required
9S before.
2 8 . In like manner, by means of ^Jingle and a triple
Line of Numbers, two mean Proportionals may be
cafily found between any two Numbers, as 2 and
54 ; thus ; fet 2 on one Line to 2 on the other, then
againft 54 on the triple Line is 6 on the fingle one,
which is the Jirjt Mean 9 then fet 2 on the triple
Line to 6 on tht^ngle one, and againft 54 is 1 8, the
fecond Mean, on the fingle Line ; fo the four Num-
bers are 2 : 6 :: 18 : 54. And thus you may find
two Means between any other two Numbers, which
-in many Cafes is a moft ufeful Problem.
29. In thf foregoing Qperation? } have^made no
jl)enti9n oi Decimals, becaufe they are to be refpeded
Chap. IX. ^;^ Instruments* 167
as Whole Numbers in the Management of them by
Inftrumnial Operations ^ in the fame manner as they
yrtrt by Numerical Logarithms i the Number of^->
£imal Places in any ProduSl^ ^oHents Power^ Root^
&c. being always determined here, as in all other
Methods of working them, by the Rules proper to
Decimal Arithmetic.
30. Thus it appears what Similarity^ CobereHei,^
and mutual Relation there is between the foregoing
Methods of folvlng Arithmetical ^eftions by Loga-
rithms, both Numerical and Inftrumental ; and that
they are all one in Nature, and differ only in the
Modus operandi^ or Manner and Form of working.
By this Chapter, I prefume, 'twill be eafy for the
Learner to obferve how any common Queftion in
Arithmetic^ or the Menfuration of Artificers Work^
as Joinery^ Mafonry^ Carpentry^ Paintings Ttmber-
MeafurCj Gauging ^ &c. may be moft readily per-
formed by the Line of NumberSy with Compaflcs, or
by the Sliding- Rule ^ which is much the beft Way.
CHAP. X.
Sbeweth the Analogies ^r pRopoRTioNsy^r
the Solution of all (he Cases g^ Plain and
Spherical Triangles, both Right and Ob^
[ liquc angled.
1, TF any defire to be throughly inftrudcd ih the
X ^eory of Plain and Spherical Trigonometry^ I
muft refer them to my Toung Trigonometer^s Guide ;
fince all I intend here is only to ftiew ihegreal and
moft excellent Vfe of Logarithms in the praftical Re-
fojution of Plain and Spherical Triangles. A Sy-
nopHs
umw
\%
ain Tri-
i:
-1'^.
chap. X. and Spherical Tkiaj^gles. * r69i
2. A Synopfifi of the Analogies for the Solution of
all the Cales of Oblique-angled plaiii Triangles.
H- 1
<
1— 1
t-H
I—I
1— »
S3JBP
n > >
w n a
oS^
>n^
USAJO
n .w >
fe-">
.Sfe
on
jilSnog
AB : AC + CB :: AC— CB : AD=fDB.
In the firft Triangle, 'tis AD— DB l . p
in the fecond Triangle, AD+DB 5
Then AB — AG i= GB = 2DB. Thus each Tri-
angle is reduced to two Right ones, viz. ADC,
and BDC ; in either of which two Sides are
known whence the Angle C may be eafily found.
n
« Q
n
3 w
• ••
-I
• • a.
• • • •
c. >
On
• •
•
1
sC : AB :: sB : AC
sC : AB :: sA : BC.
■ \ ^
"^ft
;■ ■y>n
yjo Solution ofPUin
« A Synopfis of the Canons^ and the Analones (formed firom them
Ae Solution of a^Jpfte Cafes of Right-angled Spherical Triangles.
OW
now
rt> CO
XX'
W CA •
1^ M «
OOCd
»« •• ••
<^ <^ Q
rt Oi 05
J«?^^
8 8 8
now
< ^
on
SJWO
»^ »^ *"
:: ••: ::
CO CO O)
txin>
onn
5iOJ»}«
060
owS
CO CO
>>
wo
09 M
WO
wo
• • • •
ow
, wo
ca oi M
• • •• CO
:: :: -
Q
WOCfl
pws,
o
w^
"ow
M 0» M
OWQ
W W»5j
2"w|
owo
n
o ..
•• CO
•• ••
• ■ • •
wWq
p>>
o.w
»S
CO 09 . .
QWW
OWil
Xno
09 CO
OWw
www
&
M 09 CA
.^oo
• • •• ••
J2nSLco
o
w
"S
*oo
tflW
03 GO
DW
w>
• • r-r
WW
fK}pip3
fe??
09 fT tZL
%%,^
o
o
sapo
U3A1£)
jqSnos
P
P
09
>
I
o
8
OQ
• ie-S^dsfTiLflT'otn <^ V« » •:^» •:^» *xL* «:^* •vy»f
h2. •
^* ■" iji -
•^»
172 SolutiohofPIainTKiAViQhEs.
5. Being thus fiirniflied with Proportions, we
ihall foo|i fee with what incomparable Pleafure and
Eafe the fcveral Cafes of Triangles before going are
refolved by the Canon of Logarithmical Sines^ Tan*
genfSf and Secants ; and alfo by the Line ofl^umbers^
both with the Compajfes and by the Sliding^Rule ; I
fey, we ftiall fee in the two next Chapters, with
how much more Eafe and Pleafure they are re-
folved by thefe two Methods^ above what is attain^
able by any other Way yet invented for this Pur«
pofc.
CHAP,
Chap. XL Solutioff of Plain Triangles. 173
CHAP. XL
5^^ Solution ^Plain Triangles ly the
Canon ^Logarithmical Sines and
Tangents ; iy Gunter's Scale and
Compasses s and ^ the Sliding Rule.
X. T Have already defcribed and taught the Ufe of
J^ theXogarithmic Canon, fo far as to find the
I^ogarithms of any Number j Sine^ Tangent ^ qx Secant
propofed, I fliall therefore here only obferve, that
what I call Gunter^s Scale is fuch a Plane Scale as hath
upon it Gunner's Une of Numbers^ and of Artificial
Sines and Tangents j whofe Nature, Conftrudipn,
and Defign, have been before difcourfed of in the
Theory. The Sliding-Rule has the fame Lines,
which are contrived to Aide by one another as you
pleafe ; and to avoid Repetitions, I fli^U call the
Line of Numbers on the Rule it felf, A ; and that
on the Slider, B ; alfo I fhall call the Lines of Sines
and Tangents on the Rule Sr, Tr ; and thofe on the
Slider Ss, and Ts. You muft know alfo that the
End of each Line marked 10, 90, 45, is here called
Radius. Having premifed thefe things, we proccei
immediately to the Solution of
2. Right-angled Plain Triangles,
C^fe L In the Right-angled Triangle ABC, there is
given the Bafe A3=285, and the Angle a$
Bafe B=:32^ 48' 5 to find the Perpendicular
AC, and the Hypothenufe BC. The Hy*
poth. Radius, Fig. I.
The Analogy for AC is, sC : AB :; sB : AC.
174 So/uffon of Plain Triaijglies. '
In Words ;
As thcSincof the Angle C=57° ii^^g.gij^gyz
h to the given Side or Bafe AB=2 8 5=2 .454845 1 , •
SoistheSmeofthe Ang, 3=32^ 48^=9.733765 J^^^
12.18S610
To the Perpendicular AC=i 83,67=2.264038
The Analogy for EC is, sB : AC :: R : BC
In Words ;
Asthe Sine of , • . B=32^ 48'=9.733765
is to the Perpendicular AC=i 83,67=2.264038
So is Radius • . . . 90^ oo'=io.oooooo
to the Hypothenufe. . . BC=339,o6=2.530273
^. By Scale and Compajfes. To find AC
Set one Foot of the G)nipaflres to 32^ 48'=B, in
Line of Sines, and extend the other upwards to
57^ i2'=5:C 5 the fame Extent will reach from ^^5
s=AB downwards in thpLinc of Numbers to 183,67
=sAC, the Perpendicular fought.
To find BC.
Extend the CompaiTes from 32^ 48'=B to Radius
90^ in the Line of Sines ; the fame Extent will reach
(in the Line of Numbers) from i83,67=AC, to
339,o6;=:BC, the Hypothenufe fought,
4. By}he SMng'Rule. "to find AC.
Make the IJne of Numbers Aide by the Line of
y Numbers, then will the Lines of Sines Aide by each
other, and alfothe Line of Tangents by the Line of
Tangents. Having thus prepared the Rule; fet
Chap. XI. Solution of PlainTKiAnGLn^l iyj^
57^ 12' on Ss to 32^ 48' on Sr ; then againft 285
on Bis 183,67 on A, the Perpendicular fought.
To find BC.
Set 32^ 48' on Ss to Radius 90^ on Sr, then a-
gainft 183,67 on B is 339,06 on A, which is the
Hypothenufe required.
5. Cafe II. 'Given the Angles 8=32^ 48', and
C=5s57° 12^ and the Side AC=i83,67 •, to
find AB, and BC.
The Bafe made Radius. Fig. II.
As the Tangent of the Ang. 8=32^ 48'=9.8o9 195
is to the Side .... AC=i83,67t=2.26403g
So is Radius .... 90^ 00^=10.000000
to the Side or Bafe . . . AB==2 85=2 .454845
To find BC.
As Radius .... 90° 00'= 10.
is to the Side .... AB=285= 2.454845
Sois theSecantof theAngleB=:32^ 48^=10.075428
to the Side or Hypoth. BC=:339,o6=s 2.530273
NoUj I have wrought this laft in Secants for Variety
fake, and that the Reader may fee the Conclu-*
fions are the fame every Way. But this Cafe is
much better refolved by making BC Radius, as
in Fig. I.
6. By the Plane Scale and Compajfes.
As there is no ^Line of Artificial Secants on th^
Scale or Sliding-Rule (as being ufelcfs) fo this Caf?
will be beft performed Inftrumentally by the Analogies
of Fig. I. where BCis made Radius.
Thcrqfo.«^
1 76 Solution (f Plain Tr i an gl e s J
Therefore, extend the Compalfes from 32^ 48' to
57^ 12' in the Line of Sines, the fame Extent will
reach from 183,76 to 285 in the Line of Numbers 5
thus ^85=AB, the Side required.
Oi: thus. By the firft Analogy of this Cafe of
Fig. II. extend the Compafles from 32^ 48' to Ra-
dius 45 in the Line of Tangents, the fame will reach
from 183,67 to 285 =AB (as before; in the Line of
Numbers.
To find BC.
Extend the Compaffes from 32^ 48' to Radius 90
in the Line of Sines, the fame will reach from 183,67
to 339,o6=BCi in the Line of Numbers.
7. By the Sliding^Ruk.
To find AB.
Set 32^ 48' on Ts to Radius 45^ on Tr 5 then a-
g^ft 183,67 on B is 285s=AB, on A.
To find BC.
Set 32^ 48' on Ss to Radius 90^ on Sr ; then a-,
gainft 183,67 on B is 339,06=60, on A.
8, "Cafe in. Given the Angles B=32° 48' and C=:
Sj^ 12', andtheHypothenufeBC=339,96i
to find the Sides AB and AC.
The Hypothemife made Radius. Fig. I.
As Radius .... 90° 0(y.=io.
is to the Hypothenufe BCc=339,o6=2. 530275
So is the Sine of the Angle €=57^ i2'=9.924572
' 'J ' ' '
To the Side or Bafe - . . . AB=:2 85=2.454845
And fois theSine of the Ang. B=32° 4-^'^9'7 337^5
. to the Side . . . • AC=i 83,67=2^264038
Or
chap. XI. Solution of Plain Tr i an ol es. 177
Or thus; AB made Radius, Fig. II. To find AC
As Radius .... ' 90^ 00=10.
is to the Side .... AB=:2 8 5=2 .454845
So is the Tangent of ... . 3=32° 48^=9.809193
to the Side .... AC=fi 83,67:^=2.264038
9. - By the Scale and Compaffes.
Extend the Compafles frcwii Radius 90 to 57^ 12'
in the Line of Sines, the fame Extent will reach from
339,06 to 285=AB, in the Line of Numbers.
And then again, extend them from 90 to 32^48^
in the Line of Sines, the fame will reach from 339,06
to i83,67=AC, in the Line of Numbers.
10. By the Sliding-Rule.
This Cafe may be folved by once fetting the Rule^
thus ; Make the Line of Sines to Aide by the Line
of Numbers: Then fct Radius 90 on Ss to 339,06
on A J thus aeainft S7^ i^' ^^ Ss you/ee285=AB,
andagainft 32^ 48' is;i83,67=AC, on A. Such is
the great Conveniency of this fmall Inftrument.
11. Cafe IV. Given the two Sides, AB=285 and
AC=i 83,67; to find the SideBC, and the
Angles B and C, 9
The Bafe made Radius, Fig. 11. To find B,
As the Side .... AB=2 8 5=^2 .454845
is to Radius lO-
So is the Side AC=ii83,67=2.264038
To th^ Tangent of the AngleB=32^ 48'=9-8o9i93
Qx thus, making AC Radius, Fig. IH. To find C.
A a As
178 Sofutm^ Plain TmAiiQhBs.
Asthc Side AC=;i 8 3*67=2,2 64038
is to Radius lo.
So is the Side AB=:2 8 5=2 454845
' ' t
To the Twig, of the Angle C=57^48'=io.i^o7Q7
12. To find the Side BC,
This may be done direftly with the Secants, or by
finding the Angles firft, with the Sines ; as is mani^
feft mm the Synopiis. But fince tf the required Side
BC be nude Radius^ it can't be found by the com-
mon Analogies of Trigonomelrj ; 1 have given an
i^quattoii torjoed on t he Principles of Geometry for
that putpofe, viz: VAfiq-f-ACqasBC i, which I
ihall refolv^ by Logarithn^ as follows.
The Log. of the Side . f . AB=:2 85=2 .454845
the Pouble thereof) .t, o,^^ ^
is the Log. of p ' ' ABq=8i225=4.90969Q
Akb the Log. of the Side AC=:i 83^67=2.264038
Add the Square • , , . * ABq=8i225
'"^^thJS*^} ABq+AC(|=ii4?59.65=5.o60546
L» .1 ,' I. I I I
Hal f which is the Log. of
i^ABq4rACq=BC=:j39,o6=s2,5jo273
the Side required, ^ -
1 3. By Scale and Com^ajfes. To find the Ang. B.
Extend the Cdmpalfcs from 285 to 183,67 in the
Line of NiMnbers^ the fame will reach from Radius
45^ to 32^ 48', inthe LineofTahgents, th0 Angle
B required* •
By
chap. XI. SfiktfOft ^ Plain Tft i A N GL & s. 179
#
By the $lidiHg'Rule.
Set 285 on A to 183,67 on B, then againft Radius
45^ on Tti IS 32° 48'=: the Angle B, on Ts.
Having thud found the Angles^ the Side BC 19
found as in the foregoing Cafes.
14. Cafe V. Given the Hypothenufe BC=:339,g6
and the Side ABe=:285 ; to find the Angles
B and C, and the Side AC.
To find the Angle C* Fig. I. BC Radius. •
As the Side * . * * * BC=3 39,05=2.530275
is to Radius ' 10.
So is the Side .... 4 ABral 8 5=2 .454845
To the Sine of the Angle Csezgy^ 1 2'sr:9.924572
To find the Side AC.
As Radius ...««,« to,
is to the Side ... * BC=a233Q,o6=2.53027-3
So is the Sine of the Angle Bsas32^ 4^ '==9733765
fcl II- ■ ■■■■Mil ■>
to the Side ./.*.* AC:2= 183,67=2*264038
Nvfe^ The Side AC may be foupdGeotneMcaify^
as taught in A rt. 12* the Equation being
•BCq--ABq=:AC.
tg. By Scale and Compajfes. To find the Angle C.
Extend the CompafTes from 339,06 to 285 in the
Line of Nunibers, the fame will reach from Radius
90^ to $y^ 12'BaC, in the Line of Sines.
By the Sliding' Ruk.
Set 339,06 on A to 285 on B, thep agalnfl Radius
90^ on Sr is 57^ 12WC, on Ss.
A a z The
%
1 8o Solution of Plain T R i an gl es.
The Angles being thus found, the Side Xc may
be found by Scale or Sliding- Rule ^ as before.
i6. Cafe VI. Given the Side AC=i83,67 and BC
s=s339,o6 ; to find the Angles B, C, and the
Side AB.
As this Cafe is, in the Nature of it, the fame as
the laft, fo the Solution is in all refpe&s the fame, and
needs not be repeated.
ij. Of Oblique-angled Plain triangles.
Cafe L There is given the Angles C=82^ 30',
B=6o^ 00' ; and the Side AB=365 ; to
find the other two Sides AC and BC,
Note^ The firft Triangle in the Synopfis of Oblique
Plain Triangles is that which I have regard to
here, and is acute-angled ; the 'D'lSttttict be-
tween this and the obtufe-angled one, Fig, II.
will be taken notice of as I go along.
To find the Side AC. Fig. I. Co. Ar .
As the Sine of the Angle C=82^ 3o'=o.oo3732
is to the Side AB=:265°=2. 562293
So is the Sine of the Angle 8=60^ oo'=i:9.93753 1
to the Side fought .... AC=3 18,82=2.503556
To find the Side BC. Co. Ar.
As the Sine of the Angle Ci=82° 30^=0,003732
is to the Side AB=365=2.562293
SoistheSineof the Angle A=37^ 30^=9.784447
to the Side required . , . •BC=224, 11=2.350472
18,
chap. XI. Solution of Plain Tr i an g l e s. i 8 i
18. By the Scale and Compajfes. To find AC.
Extend the Compafles from 82° 30' to 60^ in the
Line of Sines, and the fame Extent will reach from
365 to 318,82 in the Line of Numbers 5 therefore
3i8,82=AC.
To find BC.
Extend the Compafles from 82® 30' to 37^ 30' in
the Line of Sines, the fame will reach from {6^ to
224, 1 i=6C, in the Line of Numbers.
19. V By the Sliding-Rule.
To find AC.
Set 82^ 30' on Sr to 60^ 00' on Ss ; then agafnfi:
365 on A, you have 3 1 8,82= AC, on B.
To find BC.
Set 82'' 30' on Sr to 37° 30' on Ss ; then againfl:
365 on A, you have 224,1 i=BC, on B.
But fince alHhe Angles are known, both the un-
known Sides are found at once fetting the {lule thus ;
Let the Line oi Numbers Aide by the Une of Sines ;
and fet 182^.30' on Sr to 365 on B, then againft
60^00' is 3i8,82=AC ; and againft 37^30^ is
224, 1 r=BC, on the Line of Numbers.
20. Cafe n. Given two Sides, AC=3i8,82 and
BC=s2 24,11; arid the oppofite Angle A=
37° 30' •, to find the Angle B, and the other
Side AB.
To find the Angle B- Co. Ar.
. As the Side ...... BC=224,ii=.7.649528
is to the Sine of the Angle A=37^ 3o'=9.784447
So is the Side AC=3 18,82=2.503556
50 the Sine pf the Angle . . B=;6o^ Qofz^^.g^jSi^
The
l8i * Solution of Plain ThiAngIeS.
The Angles being thus known, the Side A6 U
found as were the Sides AC and BC in the foregoing
Cafe.
21. By the Scale and Compt^es,
To find the Angle B.
Extend the Compares from 224,11 to jtS,S2 in
the Line of Numbers, the fame Extent will reach
from 37^ 30' to 6qP oo's=?B, on the Line of Sines.
By the Sh'ding-Rulc.
Set 224,11 on A to 318,82 on B, then agamft
37^ 30' on Sr is 60° 00' acB, the Angle fought,
t>\\ Ss.
Noie^ In this Cafe of the Obtufe-angled Triangle,
Fig, II. 'tis obvious the Angle here found is the
outward Angle CBG,whofe Complement there*
fore to 180 Degrees is equal to the internal ob*'
tufe Angle ABC=i2o^ oo^
22. Cafe III. Given two Sides AC » 318,82 and
BC 224,11 ; and the included Angle C=*
82^ 30^ i to find the Angles A, B, and the
Side AB.
To find the- Angles A and B.
The Sum of the given Sides is AC+BC=tt542,93 ;
their Difference is AC — ^BCsx:94,7i •, the Sum of
the unknown Angles A4'Ba«97^ 30' j therefore
the half Sum is ^ ^ 48^ 45' S whence by the A-
nalogy in the Sypopfi^, .6y 5 . . . .
Ai
Chap. XI. SoJution of PlainTjtiAKQLES. 183
*".£'SM« "'}• • • AC+BC=54..93-.7-'«i5^^6
is to their Difference AC-^BCrs 94,71=1.976396
So is the Tangent of 7 . ^
the half Sum of the J ^ =48^ 45^=10.05701 2
unknown Angles 3
Then to the half Sum of the Angles 48° 45'
add the half IXfference , . . . 1 1° 15'
The Sum is the greater Angle B=6o° 00'
But the PiflF. is the leffer Ang. A=3f 30'
?5. By Stale and Compafes,
To find the Half PifFerence of the Angles ^— ?,
2
Having prepared the Work «fi above, pcoceed thus s
Extend the Compares from the Sum of the Si^es
542,93 to their Difference 04^71 <m the Line of
lumbers ; *with this Extentifet one Foot of theCom-
pafies in Raditxs 45^, and pitch the other downwards
in the Line of Tangents, where fix it while you faring
the other Foot from 45^ to 48° 45^5 then with this
Extent apply one Foot in 45^, the other will tieach
to 1 1^ ly =5 thcL half Ji^ifference of the Angles A and
jB ; wl^ich dierefore may fae found as befbr^.
Or if vou have a Line of Tangents coistiiiued ber
yond 45^9 th)Mi the Extent from 94,71 to 542,93 in
^the Line, of Numbers will reach from 48^ 45' to
f 1^ 15' in the Tangent Lit\e.
184 Solution (f Plain Tri angItEs,
24. By the SUding-Rule.^
Stt 54-2.03 on A to 94»7i on B ; then againft 45^
on Tr obferve the Degree and Minute on Ts, and
bring that Point to 48° 45' on Tr i then againft 45*^
on Tr you have 11° 15' on Ts, which is the half ,
Difference as before, of the two enquired Angles
A and B.
Or thus, if the Line of Tangents be continued on
the Slider beyond 45° ; fet 542,93 on A to 48^ 45'
pn Ts, then againft 94,71 on A is 1 1^ 15' on Ts.
Having therefore the Half Sum^ and Half Diffe-
rence of thofe Angles, they are found as in Art. 22.
and then the Side AB will be found to be 2^5 as in
Cafe I. hereof,
25. Cafe IV. Given all three Sides AB=365, AC=
318,82, and CB=224,ii, to find the
Angles.
In order to this 'twill be neccflary to reduce the
©blique Triangle into two Right-angled ones ADC,
and BDC, thus ; Find the Sum of any two Sides
AC4-CB=»542,93 •, and their Difference AC — CB
5=::94,7i 5 efteeming the other Side AB the Bafe, the
Difference of whofe Segments AD — DBsasAG, is
firft of all to be found by this Proportion, viz^
Co. AjT.
As the Bafe AB:^365=.7.437707
is to the Sum of I * ^ . o^ ' ' ^ ^
the two Sides J AC+BC=542,93'=2.734744
So the DifF. of the J .^ nr^ . ^, . «^^o«<:
two Sides I AC^BC«94>7i=^.976396
J- .
Thereforo
Chzp.XL Solutim of Plain Tkiasoles. 1^5
Therefore AB— AG=224=GB=2BD ; there-
fore 5H==:ii2i and AD=253 ; and fo the whole
-T^^gleACB is reduced to two Right-angled ones
Apfc ancJ*BDC», in each of which there is two Sides
IJSlD and.4X^ DB and BC 5 by which means
jles ACD^'and'DCB may be found, which
_ 3er arfe iB^ual to" the Angle ACB ; and this be-
^Itnoyn^-Ullye other two are found with Eafcby
iJaft^ Andtliusth^ whole oblique Triangle isre-
fiailv^d.
l^ie^ When the Perpendicular falls without, as in the
obtufe-angled Triangle ABC, Fig. II. then \t
will be thQ Sum of the Segnlents AD+DB—
AG ; and the Difference of the Angles aCD—
BCD= ACB, the Angle required.
26. And fince the feveral Problems ofNavigaticn^
whether in the Plain^ Mercator*s^ Middle Latitude^
Oblique^ or Traverfe Sailing ; as alf^ of meafuring
Heights and 'Diftancesj accejjible and ^inacceffihle ; of
Fortification^ Gunnery ^ and divers F^rtf.of ^ronon^^
&c. are all refolved by the Doifrine of Plain Trigo-
nometryj as in the Method before-going ; it muft be
very eafy for any who underftands the Solution of
Pkin 'Triangles, to apply it to any praftical Cafes
that may occur in any of the aforefaid Arts, wirt-
out any farther Inftruftions or Examples. Yet thofe
who would fee the Theoty of Plain Trigonometry, and
its application J in thelargeft Extent, may fitid it in
the firft Vol. of my Toung Trigonometer*s Guide,,
B b C H A K
i86 SobtmdfSphfricalT^tAnoLs^
CHAP. XII.
5i&^ Solution (jTSPHERicALTiliANetES fy
Logarithms, Ay Cunter'Jt ^cAle^ dm
by the Slidino Rule.
1. A S in the fbregoing Chapter I have exempli*
Xx fied the Refolutjon of Tlain '^ianl^s bbl^
hy tht Canon of Logarithms^ and l^arilhmc'al In-
Jirumcnts^ fo I fliall puriue the lame Med)od here
with refpcft to Spberkal Triangles of b6tti kinds \
in each of which there are Jffx diff'erent X^afes^ which
in all their Variefiis are refolvable fording to the
jinabgies ^llignred in the Synapfis % o^ which^ and on
the Figure of the Triangle thetr, the Reader is dc-
fired, to have his eye, thro* the who|c Courfe of Ex?
ampler. To begin then wjth .
2 . ^ kigbt-angled Spherical TridngteSi
Cafe I. Given the Bafe Al^=38*^ 15^, and Ahgfeiit
BafeB==39^ 5& to find the Perpendictihr
AC, the |lypothenufefiC, and the AnglpC.
f . To find thp Pcrpendioiikr AC.
As Radius ..... 90^ oo'=i 0.00000a
is totheSineoftheSide ABzzjiS^ i^'= 9^791757
So is the Tang, of the Ang. 8=39^ s^'^9:9'^'^7^7
to the Tang, of the Perp, AC=27^ 23'j=9.7i^54^
d. To find the HypQthfanufc BC.
As Radios ....... 90° oo'=i6.ooooo6.
to the Co-Sine of the Angle 6=39° s&zzL^M^G'jy
$0 i$ t^tOa-T. of thcSidf AB=38<5 i5'=:io.i032jB8
to theCo-T. of the Hypoth. 80=45^ 48'=:9.988o65
3. To find the Angle G.
A$ JRgdjiu? . . ^ . . . . . 90^ oo'=:i6.ooooo6
t;Q 4c<^<p-Smfi of .the Side ABrsjS^ 1 5'=9.895p45
., §9 13 .the Sipe of , the Angle 8=^9^ 56'=^. 807465
to the Co^Sine of the Angle €=59® 44^=9.702510
3. '^jr ^^7/ and Compaffesi
I. To find AC.
. Extend the Co^npafles froni ^o^ to 39^ $& in the
Line of Sines, that will reach from 38"^ 15^ in the
jLioci^of J^ngeiits ttp 27^ 23'j=AQ the Side re-
J5[uircd.
2. To find the Hypoth. BC.
Extend the Compafies frooi Radius 90^ to the
^Po-Siie. ,Qf ^B, 50^ 04' in the Line of -Sines •, then
.^pply th^t Extent from 45^ in the Line of Tangepts
down Wards, ^hc^eiix that lower Foot, and bring
4he.other-to 51^45' the Co- Tangent of AB^ this
Jaft Extent. will reach from 45^ to 44^ 11' thc,Co^
Tangent of BC=45^ 4^'* ^^^ ^^^ required.
3. To find the Angle C.
Extend the CompatTes from 90^ to 51^ 45' the
Co- Sine of AB, the fame will reach from the ^ino
^39"^ 5^' to 30^ i& the Co-Sine of 0=59^ 4V.
Bb 2 4*
i88 Solution of Spherical Tki ah gles.
4- By the SUding^RuU.
1. To find AC.
Set 90^ on Sr to 38® 15' on Ss; then againft
39° 56' on Tr i$ 27° 23'J=AC, on Ts.
2. To find BC.
Set 90 on Sr to 50° 04' the Co-Sine of B, on Ss ;
and mark the Degree and Minute on Ts againft 4^^
onTR, bringthat Point to 51^45' (the Co-Tangent
of AB) onTR, on which againft 45^, you have
44^ 12' on Ts, the Co-Tangent of BC==45^ 48^, as
required.
3. To find the Angle C.
Set 90^ on Sr to the Co-Sine of AB, 51° 45' on
Ss ; then againft 39^ 56 on Sr is 30^ 16' on Ss, the
Co-Sine ofC=:59^44', as required.
5 . Cafe II. (ji ven the Perpendicular AC=2 7^ 2 3', and
the oppofite Angle B=39° s^' » ^o fi^d the
Side AB, the Hy pothcnufe Bp, and Angle C.
TofiidAB.
As the Tangent of 6=39^ 56^=9.922787
IS to the Tangent of • • AC=27^ 23'i=97i4544
So is Radius .... 90^ 00^=10.
■'°li^e%f=?'} AB=38».5'=979.757
The Operations of this Cafe being only thcConverfe
of the foregoing, needs no further Examples, iri
Numbers.
6.
ChXll. SpktiM of SpbericaiTKi AUQL^s. 189
6. By Scale and Compares.
1. To find AB.
Extend the Compaflcs. from 39^56' to 27^ 23^1
on the Tangents, the fame will reach from 90^ on
the Sines, to 38^ i5'=:AB.
2. To find BC
Extend the Compoffcs froifa 39^ $& to 27^ 2^%
on the Sines, the fame will reach from 90!^ to. (the
Sine of BC=). 45^ 48^, as required. *
* 3. To find the Angle C
Extend the CompafTes froni (the Co-Sine of AC)
62^ 36^1 ^d 90^, the fame will reach from. (the Co-
Sine of B) 50^ 04' to 59^ 44' the Sine of the Angle
C required.
7. ' By the Sliding' Rule.
I. To find AB.
Set 39° s^' onTR to 27^ 23'^ onTs, then againft
90^ on Sr is 38^ i5^=AB, on Ss.
2. To find RC.
Set 39^ s^' on Sr to 27^ 23^1 on Ss 5 then againft
90°.on Sr is 45^ 48'=BC, on Ss. '
3. To find the Angle C.
Set (the Go-Sine of AC) 62® 36'j on Sr to 90^ on
Ss ; then againft (the Co-Sinc of B) 50^ 04' oh Sr is
59^ 44'=^C, on Ss.
8. Cafe III. Given the Hypothenufe BC=r450 48',
and an. acute Angle B=39*' c&^ to find the
Legs AB, AC, and the Angle C.
I.
1^ $M«ii ^Spt&icalfniAtmiisi
i. To find AB.
As kadius .<*,.. .' . 90° oo'sasio.oooooc?
to the Co-Sine of . . . lis=39° 56'=?; ^M4J^yf
So is the Tangent of . < BCas^^° ^8'=io,oj2?29
As Radius ..*.••* ^d^ 6(5(is=ip,
is to the Sine of , ^ . ; ''BC*:4:5<^ 48'-? 9.8/54^^'
Sois theSin6 of • • « . B=;=39^ 56'=s9.8o7465
fought - } AC»;=17^ 2fjt^^Mm^
3. To find the Angle C. .
As the Co-Sinie of . . . .BC;=i^^ 48'=9.84333_<5
is to Radius . • . . ^ 90 00=10.
So is the Co-Tang€nt ofBii!i339^ 5&s=zio.oyy2iif
■■ - '1
Toithelfaiigentat. . . ^C;azg^^ 44^10.^3387/
9. By Scalfi ,and Qifnpajes.
^. XocfindAB.
Extend the Compflfes iwm Radius .9oP4»^(^ O^
fine of B) 50^ 04' in the Sines, then fet one Foot iii
45° in the Tangents, arid pitdhthe other downward ;,
^here &c it^ while you bring the former from 45^
to 45^48^4 then will this laft Extent reach from^j;^
to the Tangent of 38^ I5'=3 5lB^ theSidc peqqircd. '
2. To find AC.
^Extend the Compafles from 90^ to 45^ 48' in the
Sine$»
Sines, the fame will reach frofp. sg° s^' IP ??** '^3fi
SsAC, in the^kme Line.
3. To find the ^leC.
Extend the CompstfTe; from (the Qo-Sme of BC)
44° 16' to Radius ^^ j the fame applied to (the G>-
T«agent of 8)50^94', wiJl t«acji>»fh^ Tfsgent
5j>^ 44'*T^» the Angjie Mqujied.
I. T<> iiod A?.
Set Radius 9p° on Sj| to 50° 04'^ CVSbe^f B)
ion Ss, and a^inft 45° on Tr mark the point oa T«,
"bring that Point to 45° 48' on Ts. ;j and now agaiaf^
45° on
hat romt to 45^ 48' on Ts. ^ and now aeunll;
-° — Tr you hav§38° i5'=AB, tm Ts,
?.ToiSndAO.
Set R^dius^o" on S to 45* 48' on Sr, then ^gaioft
39* 56' on Se, ypu'Jl fee 27" 23'i on Sr, £hc Side
AC eegtiire^.
3, To find the Angle C.
Set (the Go-Sine of BC) 44* i6'on Ss to Radius
go* on Sr j then againft (the Co-Tan^nt of fi)
Sp'oVonTR, js59<'44'=;C, onTs. '
II. CafeJV. Given the Xep, ABz^jS' 15', an4
AC— 27' 23'j } to W the reft:
To find the Hypothenufe BC
As Radius ....,,. 90° oo'asio,
^s to the Co-Sineof . . AC=a:27''23J=9.948388
So is the Co-Sme of . . AB=38* i5'w:9.895045
•fo^he Go'Sine pf . , . , ^Csss^is" 48'ss?9.843433'
19^ Solution of Spherical Triangles;
As there is nothing new in finding the Analogies B
and Q I fhall pais them by.
12. By Scale and Compajfes^
I. To find BC.
Extend the Compafles fi-om go"" to (the Co-Sine of
AC) 62* 37'; the fame applied to (the Co-Sine of
AB) 51^45' will extend to (the Co-Sine of BC)
44'' 16' ; therefore BC=45^ 48'.
2. To find the Angle B.
^ Extend the Compafles from 38** 15' to 90^ in the
Sines, the fame will reach from 27"* 23^1 to 39*^ 56'
=8, in the Line of Tangents,
3. To find the Angle C.
. Extend the Compafles from 27° 23^1 to 90** in the
Sines ; the fame will reach from 45'' to a Point in the
Tangent- Line, where fix the Compaflb while you
bring the Leg from 45^ to 38'' 15', then will this laft
Extent reach from 4^° to 59^ 44' =C in the Tan-
gents.
13. By the Sliding' Rule,
I. To find BC,
Set 90'' on Ss to (the Co- Sine of AC) 62** 36'! on
Sr, then againft (the Co-Sine of AB) 51"* 45' on Ss
is (the Cp-Sine of BC) 44"" 16' on Sr •, wherefore
BC=r45''48', is required.
2. To find the Angle B.
Set 38*" 15' on Ss to 90° on Sr ; then againft
27"" 23^1 on Tfs is 39"" 56' on Tr, the Angle B re-
quired.
3
Ch; Xlii Siihaiwi ^-^y^^itikviGiis. i^
0; To $fid the Angle is. .
^ 27^ 23' ofi Ss to s^ b'h Sit ; mi. fiistrk the
Point on Ts agairift 45^or< X'» i bring that Point tO*
38® <5' on Tr J, thc&agaihflr 45* ofi T* Joi ft«
55^? 44'^c;'6tvT«.- ' •••..■■•••.
i4. afeV. Gf7etitheHyp6thcinjfeScd?4^'»48'.
and one $i4c ^Cata?** 2.g,'4 ; to fiQ<^^(^,rC^
Tb gnd the Angli; C.
r t
As the Tangent of A^&tj^ ^J^i^is g^ 14.5^44
Is to the do-SlAe of the hti^&i^^sQf^4/^'^^.jCii/^i$
ii . : ^jf^t^.s£^et^dJ^ V;."':
.' Extend fliA^dotni^ffesfrom45^to«5^f?i^fy^
Taiigcitisy irhen fet bne foot in 4^^ 48.^, and pitcK
the other again frOni 4,^^.4^' to 45^: thcfi (haff thrif.
laft extent reach torn HacHtts jro^ w(the Co-Sine'of
C) 30^ i& in thfe Line of Slices 1 wheitce the Angfe
Cs=59° 44^, as reijuired.
i& iStyi/ie Slidhjl'kule;'
$6t4^6xi Tr^p 27"^ ^V^5t'^^^! ahd^afk thi
fbirit inT^'ag^ntf 45^4S'*dA Tir, then brjrijf tha^
Point to 45^ on Tr ; laftly; againft-9C^oh»*; yoii
have 30^ 16' ott Ss^ w^ic^ is, theCo-Sfne of C*±y
59^44^^ asrequrrett: * ''",';'
'ATfl/i, TJh^lProportioWi M A*haMtfc:AngJc;1ft,
contain! hothtngtitw. or diflScube^r ib \Niwmrs 6i
by Inftirument> thecc^o!re.flslltrgiTe:noi £xao!ipIes' icf
them. ' .. •
1
t;. CaTe VI. Given the Angles 63=39° 56', and
Crs59° 44', to find the Sides AB, AC, BC.
I. To find AB.
As the Sine of ... ; B=39® 56'=:^.8oy4.6s
is to the Co-Sine of . . . ^=59*^ 44'=9>702452
• JSnLa Radnia acP Ort'r— Tr»
So is Radius 90^ oo'=sio.
To the Co-Sine of the ;Side ABss^Z^ «5— 9.894987
2. To find AC.
The Analogy is the fame as for AB, by which yoo
ynjl find AC=27^ 23^1 nearly.
^ 3- To find BC
As the Tangpnt of . . . €=59^ 44'=ip.2 33905
ifi to the Go-Tangeht of Bss^gl^ s&^io.oyyu^
So is Radius • . . • . 90^ 00^=10.
to the Co-Sine of the Side BC=45° 48'=. 9.843308
1 8. By Scale and Con^ajjfes.
1. To find AB.
Extend the Compaffes from 39^56' to (the Co-
Sine of C) 30^ i6\ the fame will reach from Radius
00^ to (the Co-Sine of AB) 51^ 45', in the Line of
Sines -, therefore AB=38^ 15'. In the fame mannei;
you find ACssz;^ 23'!.
2. To find BC,
"T^xtend the Cot^pafles from 59^44' to (the Co-
Tangent of B) 50*" oV in the Tangents, the famo
Extent will reach from Radius 90^ to ,the Co-Sine
of BC) 44^ 12' in the Smesj lhd-cforcBC=45^ 48'. ;
^9'
19. By the SKdif^-Ruh.
1. To find AB.
Set 39® s& on Ss to (thcCo-Sinc of Q 30' i & o^
Sr } then againft .90^ on Ss Is (the Co-Sine of AB)
51^ 45' ; confcquently AB=38^ 15',
In like manner, you find ACsssa;^ 23'!,
2. To find BC
Set M^ 44^ on Ts to (the Co-Tangent of B)
50^ 04' on Tr I then againft 90* on Sr is 44** 12'
(theCo-SineofBC)onSs. Wherefore 80=45^48^ ^
2a. Of OMique Triangles.
Oblique Spherical Tmngles admit alfo of fix difr
ferentC^fes, as follow.
Cafe I. Given two Angles, 8=34® 30^, and Dsa
48'' oo^ and an oppofite Side BCS38'' 45' ;
to find the other two Sides DC and BP ^ and
Angle Q.
Let fall a Perpendicular CA from -the unknown
Angle C to its oppofite Side BD. Then is the ob-
lique Triangle reduced to two Right ones, BAC tod
D AC. Then (ay by the firft Analogy 5 To find the
Angle C
As the Co-Sine of . ... BCsr38^ 4«'s:9.892030
^ 13 to Radius . . • • • gQ° op^ip.
So is the Co-Tangent of 62:34'' 30^=10.1 62 &6<
to the Tangent of . . BCA^ei"" 48'f=io.27a?36
Again, by the fecond Analc^y, &y }
Cc 2 / A4
1^ S(i^ipif'ffpkfr4^
Co. Ar.
AsthcCo-Siiicff , . . Bs??*"* 3p'=p.o84oo^
is to the Sine of. . . )BCA==6i^ 48'l— 9-945159
So h the Co-Sine pf . . .' 0=48"* po'=9;8255i i
fptJieSineafthp %fi^t If C Az=:^'' j,i^z=^. 3^4676
l^o?jr,finiCie the Veipendliarl^V CA felk ^vithih th©
Tnangle', ^tisplaip tKeSum of the two Angles noi^
found makes the Apgle.fopght^ viz. BGAj4-DCA=:
, ' '• ' ' ^
To find tl» Side BD^
SI. By the irft Analogy, fay ;
As Radius ...... . 50° oo'==ip.
to th« Co:Sin? of 5=34° 3° -9-9 ' 5994
So is the Tji^ht of ' . BC=s:38° 45^=9.904497
^ -die Tapgenf of the Side BAs^s" zg'=.^.i2048^
^i»^« by the Se co n d ^^laloey, % ;
i " ,' ' ' Co. Ar.
A? the l*angen> of . . . P=48' 00^= 9-954437
is to the Tangent of . . . B==34° 3p'=:9.S37i 34
fipfi the Sineiof . /^ > . AB±:|3? ^9(1=9741698
to^JPijiepir . ; J . AD=5i^° 57'=9-5332^9
Now 'ttis Evident, the Sum of the two Arche^
AB4.AD=pBD=r33- 29'-t?9° sY.-rSB^^^'* the
•fetemiulfcd: - • • ' '^^
p. ^ . ^o^4|be fye 0p. Co. Ar.
As the S^e of the Angle p=48'pQ'=:.o.i2892jr
Js ?o t^'Sineof ^ ^e 6C=338' 45'-=s9.f 9657 1
5oj?rtejSiffepf jtheAftgle ^==3^ 3o'=9:753i28
^thc Sins bfthcSide fot^t^C^S" ^o'=^.678626
Thus tbe whole Triangle is refolved according to
,»bc Data of this Cafe, .:;;.
•* ' «
il3. Cafe 11. Given two Angles Btc34^ gp', and
C=i=i07° 3P' J and u .Si4e included BG=?
38^ 45' ; to find the reft.
You let fell the Perpendicular CA, and find the
Angle BCAi=6i^ 48'^, y^ jn Art, »2.o, . Thyi, f/ott
the given Angle Cmic;^ 3p(, take the Angle BCA
=61^48'!, and there will remain fhe Angle DCA
v±45^ 41'f, by which you will find the Afigle D;=:
48^ oo^ according to the fecortd Analogy of this
Cafe. Tnejre Things being known, we may pro«
ceecl
To find DC. Co. An
Thus, as the Co-Sine of DC A2=45^ 41 '1:^0.155 821
is to the Co-Sine of BQA:^6 1 ^ 48^1=19.674 :? 29
So is the Tangeatof , . • BCzrgS^ 45 =19.^04491
To the T. of the^idc fought DC=28^ 30^=973^6^1
»4. To find BD.
From the given Angle B let fall the iPerpendiculat
BM to the udLflownfSide DC produced } then by the
firft Analogy, tnd the Angle C^^J, thusj
As Radius 90^ 00'= I p.
. to the Co-Sine of ... . BC=38^ 45'=9. 892030
So is the ttngent of 3BCK=72^ 3q'=:io.50i278
totheCo Tangept of C9>fc=:22^ Oo'j=jo.39j3o8
Then by the fecpnd Analogy, fey >
As the Cb^ine of • . DBh=56^ 3o'j=,©.'2^&2o8
is to the Co Sinepf NBC3=22° oo'|=9.967i5i
So is the Tangent of . . . fiC=38°.45'r=9.90449i
tothcT.pf theSidcfoqghfDC=r53^ 26'^ip:j29%q
19^ Solution of SpbericalTKijiVOLEsl
25. Oife III. Given two Sides BCaagS^ 45', and
CDs=:28^ 30', and an Angle oppofite to one
of them, B=34^ 30' ; to find the reft.
This Qfe being but the Convcrfc of Cafe 1. 1 fliall
not repeat the Examples.
Cafe IV. Given two Sides BC=38^ 45', and BD=
53O 26', and the Angle included 8=34^ 30' ;
to find the reft.
This Cafe contains nothing difficult if what goes
before be well underftood ; the Analogies being plain
'and cafy for the Operations, I (hall leave them to
the LeamePs Exercife ; and pafs on to
26. Cafe V. Given all the three Sides, BCsss38^ 45',
DC=28^ 30', and BDs:53^ 26' ; to find
the Angles.
To find the Angle C.
The Difference of the Legs containing the Angle
fought, is BC— CD=AM«io^ 15' 5 then ?5±^
t^iSi"" 50'h and22z±il=:2i^35'|; wherefore,
according to the Analogy for this Cafe m the Synop*
fis, proceed thus:
The Sine of the Side. . . ^€=^9^ ^s'zsg.ygSs}!
AddtheSineoftheSide DC=28^ 3o'=9.678663
The Sum is the Log. of sBCxsDC=i9.475234
Again, the Sine pfBD+AM ..^ ^a 5o'i=9722283
2
; Add the Sine of BD-AM^^^o 35'|=9.565836
TheSumis sl2+AMxs 22=AM gj,
z Z ■^ "^
Then
Ch.XII. SolutktrefSpbencalTtiiAHQLii. ipf
Then iky j
As 8BCxsDCs=:i5|,475234
is to the Square of Radius . . . Rqss26.
„ . BD-f-AM ^, BP— A M „oo,,^
So 18 8 --t ** r 8=19.288119
to the fquare Sine of $ the! c„,r-_»/, o.«bo^
Angl2ibught ' }..,SqfC=:i9.8i2885
The half whereof is .... sfC=:53° 44fs=g.go644t
Wherefore the Angle fi>u^t is CaBio;^ 28' ; or
more compendioufly thus, by the Arithmetical Q)m*
plement oftheSides BC, and DC.
27. The Sine of BCsrgB® 45'=t0.203429 Co. Ar«
The Sine of DC3s28° 3o'=.o. 32 1 337 Co. Ar,
TheSq. of Radius Rq . . s20.oo<X)Oo ... .
TheSine of £5±AM -3,0 5o'|-9.722283
The Sine of ^=^ =21° SS'k^B-S^S^S
The Sum of all . . . . • 8q|C=i9.8 12885
* 11*1 II
ThcrcfcMTC . • . . 8|C=s53^ 44^5^9.906442 .
Confe^uently . . . Cszioy9.ti'i the fame a9l)efi*e.
Having fbund one Ang)e» the others aife.'eafflf IbUod
by the former Cafes.
' jV(7/^.ByCafe I. the Angle C yfas foun^to.be
107® 30^ which ia but 2' difiercnt frofli what it 19
found by this Cafe ; whence the Reader may pbierve
the wonderful Certainty and Agreement of^the moft
difFereAt Methods of Calcubtiony and fixxoa the grjMt-
eft Diverfity oiP Data^ ' ^
, * : .. .- ^h
I V
28. The fixth Cafe Is but the fame with this, tf
the three Angles giv^n fiecfaatiged into Sides, taking
for, tl^ gr,eaJteft Aqgh^ in that Triangle and greateiit
Side in tnisj, their SuppI^ifteptK tQ, iSp.fjegpce^.
. 29. Having thys Ibiewn the bcft Methods^ both for
EkaS^filim Eafe, ofMbfving thtf fevehil Cafes of
kightJxuLOblique'angled Triangles^ I (hall leave the
Application thereof to ttie.i^noosPnbibkniS'af ^r^
i^omy^^'Gedgraphy^'Di tilling^ Ortbvdrsinies^ ^. for
^he LevPfii^'s own Rc^rtotion 5^ as^^jbg^i? it felfi y^fy
eafy, if what is here taught 6e uriderftood ; and alfo
becaufe I have both d^m^tiftrated t^e'T^^^ry, and
vcryxbfcly. applied i\ic T>9Urint oi Bpberical ^rig^;^
nometry^ in the 2d Volt of mj Toun^ ^rigmoiMtef^l
Guide^ printed for Mr. J. Noon, at the PFbiie HarU
in VUi^^ { and wB.Ch' Irecommeml^tcr all uhd<r-
quainted^i^ith, and^nthcLW<9uIdhate a good NotiM of
the nob]e;dQd moft ofeful Art otTrigmofpeify^^ -
: , C H^ p. XIIL
Merca t qj i' ^ Sailimg performed by the Ca-
non ofho^hioi i-it^H Ktre-T A N gen f^, njottbr
^f the MsAiDj'air A^ PaAts. /
l^r«^I*E*^cipferty ^f^Msf^^aidr^s tkdrt is its'hiV-
J[ ing the Degrees of Lafitwie iWereaicd-^ih the
fame Proportion as a Pegrce qf JL^n^tpdft decreafes
friifitfcfe-E^u&ft'rtb the Pole. Whtch' Ptdportibri
is tha«» of iJfliffiifif to the Co-Sm of Latitude i or, ^of
the A?rrf*^ of -tfee ^Latitude tor itii Radius '9 which id
ihus dfcmoiKffirited,^
- 2. JLet AEB (F!g;1^ift) be-a Seftor in the Plane
of the Ecjuator, made by the* Iriterf65lions of the
l^htxes of two Meridians therewith, viz. 4E, ar ^
Cb*Xni.gfLooARiT»Mie Tangents, aj?!.
BE^ whofc Inclination, or Angte BE A, that is, ^h!^,
Arch of the Equator AB is = i Degr^* Alio let
EC be the Radius of any parallel of Latitude ; thca
iftiall DC be an Arch in that Parallel fimilar to (AB)
one Degree of Longitude In the Equator, or it is
that Degree diminiihed. Now (from the Elements)
the Arch AB is to the fame diminifh'd in the given
Farallel DC, as the Radius of the Equator AE to
the Radius of that Parallel CE ; but fmce AE=EC
(AC bdngthe giVen Latitude) and bstaufe of fimilar
TrianglesEcCand ESA, therefore Ec (EA) t EC :i
ES : EA 5: AB : DC. But EC= E c, the Co-Sine
of the Latitude AC, and ES the Secant thereof y
therefore, fc?r. which was to he demonftrated*
3. And becaufe in Mercator^s ProjeSion ot Charts
the Meridians and Parallels are all reprefented by pa»
rallel RigM-Lines^ the Arch CD in every P^ralfcl is
ever equal to that in the Equator AB i therefore that
the true Proportion between Longitude and Latitude
might be preieryed on this Chart, as on the Globe it
felr, 'twas necef&ry the feveral Degrpes of Latitude
(hould be fuch as belong to thofe Circles whofe Radii
are feverally equal to the Secants of thofe Latitudes.
4. Let A =± the Arch of one Degree in any Lati-
tude ; ^, ^, r, &c. =s: the feveral enlarged Degrees
on the Meridian ; Raa Radius ; and S, s, j, C5?4
the Secants of thofe Latitudes. Then it will'
{Aift : tf :: R : S. 1 Therefore we ftiall have
A2d :^ :: R:s. igAr^+^+c:: 3R: 84*
Ajd : c t:K :sy &c. J s-j-i.
That is, the Sum of the Secants of i, 2, 3 Degrees
is always equal to theDiftance of the Parallel of thofe
three enlarged Degrees from theEquatgr. And con-
fequently by a continual Addition of the Secants of
f' i|', 2|', 31^ &c. or their Doubles i', 3', 5' 7',
(^c. the Line or Table of Meridional Parts may be
made ^ and by fome has been made in this manner
Dd ^ for
202 Me RCATOR'i Sailings by the damn ,
for every Minute erf the Quadrant; but a truer way
is that delivered in Chap. X. of the Theory, which
fee.
5. If then you draw any Line AB (Fig. X.) to-
reprefent an Arch of the Equator, as fuppofc 60 De-
grees ; and on every tenth Degree be ereded Per-
pendiculars, thefe fhall be the Meridians ; on which
if the Degrees of Latitude, enlarged in. .the Manner,
and Proportion above defcrib'd, be fct ofF; and thofe
Divifions join'd by Right-Lines, reprefenting the
Parallels of Latitude, Mercafor^s Chart will be con-
ftrufted for the given Longitude and Latitude ; and
a Meridian thus j^raduated, is what is called the
Nautical or Meridian Line. This Chart, as here
drawn, includes 60° Longitude^ and 80^ Latitude^
that the Reader may have a perfpicuous Idea thereof.
6. Now fuppofe F be the Latitude whence you
fail, and C the Latitude you arrive to, on the Rhumb
FC ; 'tis plain there will be formed a Right-angled
plain Triangle FCD, wherein FDis the enlar^d Dif'^
ference of Latitude^ and CD the true Difference of
Longitude^ and the Angle CFD the Cc«r/^ or Rhumb ;
and therefore any two of thefe being given^ the reft
are found by the Analogies of plain Triangles before-
going.
• 7. Since every Degree is equal to 60 Minutes or
Nautical Miles, let i^^f roper Difference of Latitude
be reduced to thefe Parts, and fet (off the fame Scale
as AB was laid down by) from F to E, and draw
EK parallel to DC ; fo fhall there be form'd another
Right-angled plain Triangle FEK ; in which FE is
the proper Diff^ence of Latitude ^ EK tht Departure i
KF the Dijiance failed on the Rhumb or Courfe EFK.
Any two of which Parts being given, the others are
to be found as aforefaid.
8. And thefe two Triangles FCD, FKE, com-
prehending all the Particulars of Mercatofs bailing ;
*tis obvious enough how they are all refolv*d^ either
by
Ch.XIII. gjTLoGARn^HMic Tangents. 20 j
by ProjeSlion or trigonometrical Calculation \ but my
Purpofe is here to fhew, that "^itkont tho^^ common
Methods^ they are all to be refolv'd by the Canon of
Logarithm-Tangents, only -, in order to whicR, the
following Articles hiuft be premis'd and duly ob-
ferv*d, viz. ^ ■ ^
9. Firft, That it has been (hewn, the Nautical or
Meridian Line^ or Scale of Mercaior's Cpart^ is no
other than a Scale of Logarithm-Tangents of the
Half- Complements of the Latitude. Secondly, that
fuch Logarithm-Tangents of Mr. BHgg*sForni, (or
thofe in common Ufej are a Scale of the Differences
of Longitude upon the Rhumb, which makes an
Angle of 51° 38' 9^'' with the Meridian. Thirdly,
the Differences of Longitude on differing Rhumbs,
are as the Tangents of the Angles of thofe Rhumbs
with.the Meridians ; as is evident from the Triangle'
FCD (Fig. X.) And fourthly, that the Logarithm-
Tangent of the Angle 51° 38' 9^^, viz. 16.10151Q4
is a conftant Faftor in thefe Computations. On theffe
Premifes ^tis eafy to operate the Propofitions of Mer-
cator^s Sailings as follows.
ID. In Fig. II. letF reprefenttheL/z^j^rJV Pointy
whofe Latitude North ' DF=;49^ ^p{ ; alfo let K;re-
prefent Barbadoes in Latitude North 13^ io'=CK ;
then is FE = the Difference of Latitude ; CD the
JDifference of Longitude ; and EFK the Angle of
the Rhumb, FK with the Meridian.. Laftly, let FO
be the Rhumb making an Angle of 52^ 38' g" with
the Meridian ; or that on which the Difference of
Longitude Op is the Difference of the Logarithms
of the Tangents of half the Complements of the Lati-
tude PF, PK, or PO: Then,
II. Cafe I. Given the Latitude of the Lizard^
49^ 55' N. and that olBarbado.es 13° 10' N>
and their Pifference of Longitude 53^ 00'=
D d 2 3180
io4 Me ftcAtOR'j SaiHngt iy the Catm
3180 Nautic Miles r I demand thdCourft
and nftance M'd ? ^
{TheLat. 1
CK=i3° 10
DF=49°55
•Then the Log. Tangent of { f^^^^.^
The DifFerence of which 19 . . , QP=3372,6o
(Note,^ The four firft Figures arc Inte^ers^ according
to the Theory)
<■
Then fay, for the Courfe KFE ;
As the DiiT. of Log. Tang. 00=3372,6=3.5279^5
tothegiveh I>iff. of Long. €0=3180=3.502427
.So is the cohft.Tang. of OFE=5 1 ^ 38/ 9''^= i o. i o 1 5 1 o
y ' /.
3.603937
to the Tahgent of the 7 rrr-T? o /
Courfe f?ught |KFE=49^ 59^1=1 0.0759 7^
Secondly, For the DiftanceFK.
The Difference of Latitude is 49^ 55' — 13° io'=*
36^ 45^=2205 Miles ;
Say, as Radius 10.
to the Diff". of Lat* • .. • !EF=2 205=3. 3434091
So is the Secant 1 vt:>t> o / // o
oftheCourfe { KFE=49^ 59' 10^^=10.191807
to the Diftance faii'd FKi=:3429,38=3,5352i6;
By this Propofition you eftrmate the Courfe a Ship
muft fteer, and the Diftance of her Port.
12,
Ch.XUI. ^Logarithmic Tangents. 205
12. Cafe 11. Given the Latitude of the Litard
49^ 55' N. zxiAo( Barbadoes if 10' N. and
fhe Courfe 49^ 59^, to find the DifFct^nce of
Lx)ngitude, and Diftance failed.
Things being prepared as before, fay j
As the Log; Tang, of OFE=5i^ 38' 9^'- 10. 101 510
•".S'cwf- °'} 1^^=49° «' .o'«.o.o7597.
So is the DifF. of the Log. I
Tang, of the | Cpmp. >Qp=s*3372,6^3.5ft79€5
of the Latitudes J - . . :^
3.603937
wir ifi'r^
to theDiff. ofJl40ng. required CDs3B3 1 8osa 3.502427
Which <x>n verted foto Degrees makes 53^ oq'.
Andib HHich is the Difference df Motion or Time
between thefe two Places. The Diftance on the
Rhumb will be found 3429,3S=z:FK,*as before.
13. Cafe IIL Given the Latitude of the LiTMrd
49^^ 55^^' *he Diftance failed 3429,3 8 Miles
on a Courfe 49^ 59' 10^ South wcfterly, 'tis
required to find the Latitude and Longitude
of the Place to which the SJup is arrived.
As Radius , 10.
to the Diftance failed FK=3429,38==?3. 535215
So is the Co^ Sine of J i^ft? o 1 h « «
the Courfe }KFE=49° 59' io'=9. 808 193
. to the Diffw of Latitude FE5=52 205=2^3. 343409
Thfen from the given Lat. of the Lizard 49^ 55'
Subd.; the Diff. of the Lat. 22053^36^ 45'
There remains the Lat. {ought . . , =13° lo'
And
ao6 Me r c ator V Sailings by the Canon
. And having thus obtained the Latitudes, the Dif-
fcrencc of Longitude will be found, ^fer Cafe II. to
be 3 1.80 Miles or 53 Degrees ; and fo the Operation
needs not to be repeated.
14. If it fo happen that the Ship pafles the Equa-
ior^ an4 confequently has one Latitude Nortby the
iOther 5^«/^ ; then obferve two things; firft, that
ioth the Complements of the Latitudes are to be
eftimated from the fame Pole of the World. And
therefore, fecondly ; fuppofe you fail from a Norths
em to a Southern Latitude, you muft add 90^ to the
fin'tner^ and fuhftrad the latter, from 96^; then fub-
ftracSt this. Sum and Remainder froib 180°, and take
the biiFerence of the Logarithm-Tangents of half
the Remainders, as before.
15, Cafe IV. Suppofe I failfrom Latitude 48° 30' N.
. ( to Liatitude'23^ 45' S. on a Courfe 43° 50'
. Southwefteriy ; required the 'Difference^ Lon-
i gitude^ and Dijlance failed ? -
Then 48° 30^4-90=:i28^ 30' ; and 90— 23^ 45'=3
' 66^ 15^ And
ig'o— 66^ 15^=113045'!, 556° 52^1=10.185410
'180—128^30^= 51"^ 30^^125^ ^5' — 9-683356
The DifF. of thofe Tangents is ... 5020,54
. Therefore fay, for the Difference of Longitude;
As the Log. Tangent of 51° 38' 9^^=10.101510
is to the Diff. of the Log. Tang. 5020,54=3.700750
* So is the Tang, of the Courfe 43^ 50^=9.982309
3.683059
to the Diff. of Long", required 3815,5=3.581549
That is, inPegrees,=a:63°35^|»
To
Ch.XIIL g/'LoGARiTHMic Tangents. 207
To find the Diftance faiPd 5 fay,
.As Radius , lO.
' to the Sum of the Lat. 72^ ^ 5'==43i5^3^6^6gS^
So is the Secant of the Courfe 43^ 50'= 10. 141 849
to the Dift. faird in Miles, ^oo^^s^zs'.yySS^S
16. Thus you have all. the PraSical Cafes oi
Mercatof^s Sailing performed by tYi^Qanon ofLoga--^
rkhm-Tangents only ^ without the MeridionaV Parts ot
Charts as in the common Way. And fince this' is .
themoft exaft, and natural Method of Navigation
(next to the Globular Chart it felf) and wholly refol-
vable by Logarithms^ jt adds not a* little to the (be-
fore invaluable) Eftimation :of thofe excellent Num-
bers ; and renders their Ufe to Navigators more ne-
ceflary than before.
CHAP. XIV.
0/ /i&^ Mensuration (j/SuPERFiciES and
Solid Bodies by Logarithms,
I. \ Mongft the Variety of Methods for meafuring
£\ the Surfaces and Solidity of Bodies^ I intend
here to ftiew the Excellency of that by Logarithms 5
which may juftly be allowed the Preference to all o-
thersinpoint of £^y^and JSrm/y, Advantages none
of the leaft in common Eftimation. And fince thefe
Operations confift altogether • in Multiplication 3.nd
Bivijion^ I need not here repeat them by the Injlru-
ments^ as having already largely fhewn the Man-
ner thereof in a Chapter particularly on that Head.
2.
flo8 Mti^svuATioN of Sn^Jlcses an J
2. To meafun a S(ii;ARE, Fig. XII.
The Logarithm of the Side multipKed by 2, ^vcs
the Logarithm of the Area otfuperficial Content.
Examp. Let the Side of the Square be AB=c3i,57.
Then the Logarithm of AB=3 1 ,5731= i .4992 75
which multiply by 2
The Produd is thei Aur^rr ^ ^m a '
. A«a of the Square {^®^^'^996,66«:2.998550
3. To meafure a PxtL ALL BhOGK A u^ Fig.XIII.
Th6 Sum of the Logarithms of the Length and
Breadth is the Logarithm of the Area.
Examp. The Length ABs=a4i,5 and-Breadth BC=?
To the Log.of the Length ABaK4i,5=i.6i8o48
add the Log. of the Breadth BC=3i,57«=i.499275
The Sum is the Log. } * wr^TS
oftheParalIeIog^i^^^^='^3i2,i55=3.ii7323
4. To meafure a Rhombus ABCD, Fig: XIV.
The Sum of the Logarithms of a Side and the
Perpendicular Height, is the Logarithm of the
Area.
Examp. Let the Side AB=i 5,5 j and the Perpen-
dicular BE=i3,42.
Then to the Log, of the Side AB=i5,5=ri. 190332
add the Log. of the Perpend.BE^s:! 3,42=1=: i.i 27752
The Log. of the Area of I ^^n ^^ . ^,0^0,
the Rhombus i' ==^08,01=2.318084
5.
Ch. XIV« Solids iy LoGAiSiiTUM$. 209
5. Tomeafure a Rhomboides, ABCD, Pig.XV,
The Sum of the Logarithm of the longeft Side^
and perpendicular Height is the Logarithm of the
Area.
kxatn. Let the longed Side AB=:i9,5^ and per*
pendicular Height BE^6,o7
Then to the Log. of the Side AB=:ii9. 5=1. 2960^5
Add the Log. of the Perpend. BEii^6,o7=o. 783 1 89
Log.ofthe Areaofthei ^,,h ^r:. ^ r^^^r^^A
[lomboides } ==118,365=2.073224
The
Rhomboides
6. To mafare j Tri a nom ABC, Fig. X Vl.
The Sum of the Logarithm of the Bafe, and of
half the perpendicular Height (or vice verfd) is the
Logarithm of the Area.
Exam. Let the tifeife ]ic==:65,25 5 and the per*-
pendicular Height AG=2i,5 j then f AGssio./f.
Therefore,
To the Log. of theBafe . . . BC==:65,25=t.8i45?d
Add theLog.of half thelPcrp. fAG=r: 10,75=1 .03 1 408
The Log. of the Area . . . .=4=751,4375=5=2.845988
7. To meafwri tf Cjrole ABCD. Fig. XVU.
f D=:the Diameter.
In order to this, let] Pacthe Periphery.
C A:t3the Area.
Then the Rule^ or Theorems for finding thofe ft-
veral Parts are as follows ;
E t Thco,
2IO' MENstritATiOR of Svperfictes and
Theo. I. 3.i4i6D=P. TJi. II.o.7854DD=A.
TheoJII. o.3t8?P =3>. Th.IV.o. Q7957PP.= »A.
Thco. Wi.2732A=D. Th.VI.i/i2.5664A=P.
8. Therefore the Diameter AB being given, fup.-
pofc =s 20,15, to find the Periphery P? per
Thcor. I! ^
TheLog. of the Periphery P=:63,'303,6?r.=:i. 801426
9. Given the Periphery of a Circle P=63,303, f^r.
- to find the Diameter Tii per Thcor. IIL
The Log. of the Dianxcter D or AB=;20. 1 5= i . 3042 63
10. Having the Diameter given, fiappo{e=2di5,
to find the iSirp, A of the Circle ? per Theor. 11. .
The Log. of the Diameter . . AB=20. 15=1.304275
Multiply by . 1.. ................... , _ %
The Produd is the Log. of DD=2.6o855o
II.
Ch. XIV. SoM iy hoGAtiiTHUSr ait
11. Or thus fuppofing the Periphery given,=63,303,
£g*f. per Theor. IV.
To twice the Log. of P=63.303={ [sVifze
Add the Log. of the conftant j ^^ ^^^^^^^^
Number >
TheSumisthcLog.ofj^_ 8 8 ^^_2 6oj
the Area ^o ^ ^'
12. Tomeafurethe Ssctor. of a Circle ACB,
F/ff. XVIII.
The Sum of the Logarithms of the Radius^ and
f the^rfi& (or of the Arch and half the Radius) is
the Logarithm of the Jrea.
Examp. Suppofe the Radius AC=i2.36, and the
Atch AB IP, 1 2 i then fAB=5,ii.
Then to the Log. of Radius AC=i 2, 36=1 .092018;
Add the Log. of fAB=5, 11 =0.7075 70
^ of d2'sl'?! ."^I AC'^«3.^59«=r.7995»8
i». To meafure the Sbpm?nt of a Circle, «8
AFBG, Fig. XIX.
The beft way is to find the Centre C •, as by this
Theorem, ^/z. TB -FG=N. then ^ ^CG;
whence C is given ; and finding the Are» of the
whole Seftor ACBG, (as per laft Article,) and the
Area of the Triangle ABC, (per Art..60 tf th^ awer
Area be fubdofted irom the former, it wiUJeave
the Area of the Segment required, AFBG,
Ee.2 . . '. - H.
^It MEMSUitATioK of Superficies and '
14. To meafure a Spherical Triangle, ABC.
Ftg.XSi.
From the Sum of the three Apgles A, B, C, take
186 riegrees \ then from the Logarithm of the Re^
puindcrfubduft the Logarithm of the conftant Num-
ber 720 ; to that Remainder add the Logarithm of
the Superficies of the whole Sphere ; the Sum ihal|
be the Logarithm of the Area required.
' Angles be YdS ||
I
Their Sum is . • . . 190^ 45'
Subftraft ....... 180 00 Log.
There remains .... 10^ 45^= io>75 == 1.031408
}Jubdu<5l: the Logarithm of 720 = 2 J57332
8.174076
TheSumistheLog.ofthei -^^ «« /:^>, \ «>,«., ^
AreaoFtheTrSngle.. }4»^=^3.623:^i.2735i5
15. To meafiire an Ellipsis, as ACBD, Fig.XKl.
The Sum of the Lo^rithms of the Tranfvcrfe
Diameter A B, the Conjugate Diameter CD, and the
tonftant Number 9.7854 •, is the Logarithiu of the
Area. " • * . r
Exam. Let AB=6i,6, and CD=44,4.
' • rAB=6i,6i=i.78958i
Then add the Logarithms of < CD=r:4 4,4=1.647383
' " ' * - tN.0.7«54=9.895O9i
^h. XIV. Solids by Logarithms. 2 j|
16. To mcafure a Parabola, as ACD, Fig. 22.
From the Sum of the Logarithms of the Bafe AB^
the perpendicular Height CD, and the Number 2,
fubftrad the Logarithm of 3 \ the Remainder is the
Logarithm of the Area.
Exam. Let AB3=6i,6, and 0)2=44,4 ; as in the
ElUpfe.
rAB=r6l,6=rl. 789581
Then add thffXogjarithms of < €0=44,4=1.647383
Land N.'>'.=o,30i030
3737994
Subftrad the Logarithm of • . 3=:o.477i2i
The Log.of the Area requiredsssi 823,36=53.260873
17. TomeafurcanyRBGu larPolygon,F/j'.XXIIL
In order to this the following Table will be verf
expedient.
^Numbers.
A Table for
the more
ready find-
ing the A-^
rea of any
Regular
Polygon.
Sides.'
Names.
's^
Pentagon.
6
Hexagon.
7
Heptagon.
8
'Odhigon.
9
Enneagon.
10
Decagon.
II
Endecagon.
12
Dodecagon.
1.72048
.2.59808
3.63896
4.82843
6.I8I83
7.69421
8.51425
9-33012
Then the Sum of double the Logarithm of the
Side of the Polygon, and of the Number in the
Table proper to it, is thp Logarithm of the Area.
Exam. Let the Pentagon ABCDE be propofed,
and let its Side be ABsi4»6«
JS14 Mensuration (f^Superficm and
Then twice the Logarithm of AB= 14., 6 i ^-^^4353
^'1 1- 1^4353
AddtheL(^.ofthe/tfi«/tfriVi«»^.i.72048=::0.235649
The Log, of the Area required =5:3^6, 74=2.564355
In the fame manner you find the Area of any o^
thcr Polygon mentioned in the Table.
18. Of the MjPNSURATION ^SOLIDS.
To meafure ^ Cube ABCDPGE, Fig. XXIV.
Three times the Logarithm of the Side is the Loga-
, rithpfi. of the Solidity.
Exam. Lf(t the Side AB ==: 31.57 = 1499^75
Multiply by 3 3
Jhe Log. of the Solidity . • . 31464,81=4.497825
19. To meafure a Parallelopipedon AD,
'" ' fig.'YiyiN. ' *"
The Sum of the Logarithms of the ^readtb^
T>eph^ and Lengthy is the Logarithm of the So-
lidity.
Exam. Let th|B Width AB=2 1,5$ ; the Length
AG=3r,57 J and Depth GFa^|p,o3,
rAG=r3;i,57=i.499275
ThenaddtheLogarithojsof<AB— 211,56=1.333649
jLGF^ 9,03=0.955688
The Log. of the Solidity J_>- r ' r 00 r
, Inquired / }=6i46,2^23=:3.7886,2
20. To meafure a ^rism ABCDEF, Fig. XXVI.
Firft find"thc Jrea of the Bafe^ wHether a Triangle^
Square, &c. Tfreii the Sum of the Logarithms of
the
Ch. XIV. Solids by Logarithms. 215
the feid Aredy and Length of the Prifmy is the Lo-
garithm of its SifUdity.
Exam. Suppofe a Prifm of a triangular JBafe^ as in
theFigure, then let its Area be ABC=70i,4375i
and its Length 80=70,15.
Then add the Loga-J ABC=7oi,4g75;=;:2.84598J
nthrpsof both . . . c BD= 70, 15 =1.846028
■ '■ *
The Log. of the Solidity zz:49205,845&?r.=4,692oi 6,
^1. TomeafureapYUAMiD ABCDE. i^;f.XXVIL
Firft find the Jrea of its Ba/e^ Whether triangular^
quadrjmgular^ &c. then the Sum of the Logarithnis
of the Area of theBafe, and 7 of th^ perpendicular
Height, is the Logarithm of the 5^//ii//y. ^
Exam: Let the Priim have a quadrangular Bafe
ABCDas:j42»25 ; and the pe'rpendicukr Height
GE=:i8oi thcniGE=5:6o. i
Therfifotre add the Lo- c ABCI>=:34z,25=2.534^3
. garithm^of. .... A ; TGE=6or=iw778i5i
The Log. of the Solidity . . . : . =205^5=4.312494
< • ^ .■• . '
' 22. To meafure a C y l i n- d e li "AGBDEF;
%.XXVIIL
r
The Sum of the Logarithnis of the Area of its
Ifefe, and of the Height^ is the Logarithm of the
Exam. Let the Area of the Bafe be 380, i5=AGBH ;
i and ;the Height BD=5o,05. ,
r . l:hen add the Loga-cAGBH=38o, 15=2.579955
t rithpasiof ...... A BD;==5o,05==i. 699404
Log, of the Solidity^ . • =i9o'26,5075=:4.279359
22.
ti6 MENStTR ATiON of Supetjichs and
22. To mcafurc a Cone AEBFD, ^.XXDC,
The Sum of the Logarithms of the Area of the
^Bafe^ and -f of the perpendicular Height, is the Lo-
garithm of the 5^//rf//y.
Exam. Let the Area of the Bafe be 100.75= AEBF;
and the perpendicular Height €0=19,95 ; and
therefore ■yCD=6,65,
Then add the Loga-c AEBf'=ioo,75=:2,oo3245
rithms of • ...... 1 TCD=:6,65=:o,82282a
The Sum is the Log. of the 1 ^^ . ^^ ■ ^ ^ -^
Solidity .;....:.. .-, . .{66^,9875=^.826067
23. To Ineafurc t^ Frustum of a Pyramid;
Pig. XXX.
If it be a iquare> D:±=Side of the greater Bafe AB.
Bafeas ABCDJ d ==Side of the leffcr Bafe ER
then put-. . . C D-~dr=y and H rsthe H eight GO,
Then we have this Theorem Dd-j-i^^ x H=: the
Solidity.
Exam. Suppofe D=50, d==2i j and H=i05,6;
tjhen D— dr=;xt=:29.
rD=5o. .=1.698970
Then add the Logarithms of jd =21 , .=1.322219
tH=io5,6=2;023664
The Logarithm of • . . . DdH=i 10880=5.044853
.III I " I I ■■
The Logarithm of ■5)«fH=296o3,2=4.47i337
To which add DdH=i 10880
The 3um is . . . Dd+i*JfxH=i46483,2*=: the SoK-
dity required.
24.
Ch, XIV. Solids by LoGAkiTHMS. 217
24. But if the Sa fes of thfe Frufttiin \it txiangu-
lar% th<? Theorem is Dd4-5^;vxo,433H=the Soli-
dity. Again, if the Bafe be any of the regular Po-
lygons ; then put N = the Number^ in Table at
Art. k 7 . proper to the Polygon, and the Theorem
will be U(l+i;ff;tf x NH^the Solidity.
25. To mcafure the Fruftum of a Right Cone.
Vig. XXXI:
Let Dcsthe Diameter of the great Bafc, AB, and
a =the Diameter CD of the leffer Bafe ; I)— d=:v,
and H=Bthe pterpendicular Height, as before ; then
the Theorem for the Solidity will be 5it±^^
or thus ^Ppiwxb.7854H=Solidity.
Eicam, Let AB=i6^ CD±=i2, and GOssp: then
D— d==x=4; andTXX==5,3': alfo Ddcsiga; and
fo Dd4"T^^=i97>3f.
Then add theLoga-C Ddil-^^Jr=i97>3'=2-295i99
^thnisof....;.] 07854=f 9.895091
C H= 9=0.954242
The Log. of the Solidity \ . =1394,8704=3.144532
2.6. To meafure a Globe or Sphere ACBD»
F;^. XXXII.
The Sum of the triple Lojgarithin of the Diameter,
and the Logarithm or the conftarit Number 0.5236
is the Logarithm of the Solidity of. the Sphere.
S£xam. Suppofe the Diamete^ of a Sphere AB=:
50,37. :
F f Then
ei8 Mensuration oj Superficies and
C AB=5o,37ss:i.702i7i
Then add the Logarithms^ 1.702172
7 I 1.702172
i o,5236=.9.7i8999
The Log. of the Solidity 1 ^-^ «,
of theVcrc f }«669i3.8«4.8255i5
27. To meafure the Superficies of a Sphere.
The Sum of Logarithm of double the Diameter^
and the conftant Number ^•1416, is the Logarithm
of the Superficies of the Sphere.
Exam,. Let the Diameter AB=50.37.
f AB=5O.37±s:i.702i72
Then add thcLogarithms of < 1.702 1 yz
t 3,14165=0.4971.51
The Log. of the Superficies . • •• 7970,76*3.901495
28. To meafure the Segment of a, Sphere,
Fig. XXXIII.
r D=The Axis or Diameter .of the Sphere CD,
Let j C=Half the Diam. of the Segment's Bafe EB.
C HsaThe Height pf the Segment ED.
Then we have the two following Theprems for find-
ing the Solidity.
. C Theor. L 3CCH+H^ xa5236=:the Solidity.
^''^' I Theor.n.3DH*— 2H^xo.5236=the.Solidity.
29. To meafure a Spheroid ACBDA,
Fig. XXXIV.
The Sum of the double Logarithm of the Leffer^
the Logarithm of the. Greater Diajtnetcr, and the
conftant
Ch. XIV. Solids by Logarithms. 419
ftmfiant tJumber 0.5236, is the Logarithm of the
iblid Content of the Spheroid.
Exam. Let the leffer Diameter CD=33 j and the
greater Diameter AB=55,
Then the double Log. of . , . Cn=33{ =J-5j85J4
The Lo^ithm of ........ , AB=55=i.74a363
The Logarithm of . . . . , ^ . . . 0,5236=9.719000.
The Log, of the Solidity . • =31361,022=4.496391
30. To mcafiire the Segmen.t of a Spheroid.
As the Soh'dity of the Sphere AFBE is to the So-
lidity of its Siegment AGK ; fo is the Solidity of
the Spheroid ACBP^ to its lilic Segment AHI.
31. Xo^ meafare a Parabolic Conoid ACBD^
'" Fig.XXXY.
The Sunx of the double Logarithm of the Dia-
pieter of the Bafe^ the lA)garithm of the Height^ and
conjiant Number 0.3927 5 i^ the Logarithm, of the
SoIidityoftheO^w^; '
'Exam. Le^ the Diameter of the Bafe AB;=r55 •, and
its Height Cp=;::33.
Then the double Log. of , . .AB^ 55 \ =1 ;7to363
The' Logarithm of . • .CD=3=33=i. 51^8514
The Logarithm of 0-3927=9. 594061^
The Logarithm of the Solidity 39201,4=4.593301,
. Ff2 32.
^20 Mensitration ofSuperficki atfS
32. Tomcafurc the Frustum of a ParabolI^
Conoid, F/jf. XXXVL *
r D=Duimeter of the greater Bafe AH.
Tothisfend, !et)d=Diameter6ftheleflerBafeCD. *
C H=the Height of the Fr#f;» FE.
Then we have this !
Theorem .
*| DD+ddxa3927H=th«Solidity,
33. To m^fure a Parabolic Spindle,
F/f.XXXVil.
The Sum of the double Logarithm of its Thick-
nefs, th^ Logarithm of its Length, and the Loga-
rithm of the eonftatit Nu^aber 0.4185.8, ^ t^e Lo-
garithm of the Solidity of the Spindle.
Exam. Let the Diameter of its greateft Circle^ or
Thicknefs ABs=s43,45 j ' and the Length CDs:;
50,075.
Then the double Log. of 'AB:^43,45{ =1:637990
The Logaritlftn of CD=50,075=i. 699621
The Logarithin of ...... . .0.41888=9.622090
The Log, of the Solidity . . s=:3;^599,6=::4.5g76gi
34. Tomeafureany oftheFivE Regular Bodies,
/7^. XXXVIII.
For this purpofe the following Table is neceflary.
Names.
•^ r
Tetrahedron
•< Oftahedron
Hexahedron
IcoAhedron
Superficies.
1.73205
" Solidity. ^
0.1 1 785
>< 3.46410 ^ 0.47140 >
6.00000
8.66025
J. 00000
2.18169
Dodecahedron J ^20. 64573 ^7,66312
Exi
'am^
ph. XIV* Solids by Logarithms; ^zi^
Exam. SuppofetheSideAB=:i2, of the Icofahdron.
^ ' F/;f. XXXVIII.
The Sum of the Logarithm of the tahularNum^
^erj and double LogariAm pf the Side is the Loga-
rithm of the Superficies'; and the Sum of the Loga-
rithm of the tabular Number j and triple Logarithm
pf the Side, is the Logarithm of the Solidity.
']rhusthedouWeLo& 9^ • • • A?===^M ^J'So!?!
The Log. of the tab. Number . . 8,66o25=5?o.93753c>
The Log. o{tiieSuper^cies:sszi24y^o&S8:sp2-c^9S^92'
Again the Logarithm pf . . . AB=7][2=i.p79i8;
' , 3
3-237543
Add the Log. of the tab. Numb. 2.1816933=6.338793
The Log. of the Solidity . .=553769,9458=3.576336
3^. Thcfe Propofitions are fufEcrentfor meafuring
any Superficies or Solid Body in common UCc 5 and
the great Eafe and Concifehels of performing the
£ime by Logarithms, is abundantly manifeft from
thefe Exampks,
^
CHAP.
i22 Mensuration i^/zVi
9QOGOQPOOOC3Q gffl OQOOC3QOOOOQQ
G H A P. XV.
^he Doctrine of Mensuration applkdu
Gauging, Measuring Timber^, tf»^
Surveying; ^A^w^/Z'^ Practical Use
ef the Plain Scale ^WSlidikg Rule,
for thefe Purpofeiy is clear fy explain d.
i» TN tfic preceeditig Chapter you have the prac^
X tical Method of abfolute or general Menfuration
of the Content of Superficies and Solids^ laid down in
divers Propofitions ; my Bufinefs is here to apply
that general Doftrine to pai^ticular Ufes as thole of
Gauging y Timber - Meafure^ aod Surveying. For tho*
i there fhew'd how the Bimenfions of Bodies might
be exprefs d in Ntmbers ; yet fince thefe Numbers
reprefent divers kinds of ^antities^ as Inches, Feety
Tardsy Poles, Chains^ &c. at pleafurc, fo they are
equally fubfervient to thbfe fevoral Arts above-men-
tionM, and muft be reduced tpthofc Denominations
or Meafures \yhich axe peculiar to each. 1 ihal;i begick
therefore witK
at. Gauging.
In this Art the Dimenfions are taken in Inches,
and Decimal Parts thereof (or muft be reduced to
fuch v) and by Inches is here to be underftood folid
or cutic Inches ; and that as well in fuperficial as^
folid Meafure. For though it be improper, geome*
trically fpeaking, to afcribe Thicknefs to a Superficies^
the Gaugers always confider them as one Inch deep^
and; accordingly compute the fuperficial Content in
Gallons, 'or folid 'Meafure.
3-
fch.!5tV. to GAuGiNb. zzi
3. As Inches are the lineal Dimet^ons cfCfugers^
fo Gallons 3TC the common Qiiantities o^ their filid
Meafitrtj of liquid Subjiances efpecially ; and Bi^els
for folid dry Meafurej 2s Mah, Cohij &c. Now
the ft andard Gallons and £i^^7, as ufed in England^
are as fbllows : the Jle or Beer Gallon sstiSz folid
Inches *9 the 9^ne Gallon sac 231 ; the Corn Gall^
1=5268,^8; and the Corn Bu/hel tst 2150,42 Cubic
Inches.
4. From whence it follows, that, fuppofing the
Dimenfions of the Superficies in the foregoing Copter
taken in Inches^ the feveral Aredts^ or fuperpcial
Contents^ will by the Gauger be underftood to be ib
ttimy folid or cubic Inches ; which therefore if he di-
vide by 282, 231, 268,8, or 2150,42, the Quo-
tients will be the feveral Contends in the refpedive
Gallons or Sujbels. For injianccj in Fig. Xll. the
Side of the Square AB=:3i,57 Inches, and the
Area was therefore found (Art. 2.) to be 996^66
folid Inches.
r 282 )996,66(=3,53 Ale Gallons.
Confe- \ 231 )996,66(=:4,3i Wine Gallons,
ijuentlyi 268,8 )996,66(=3,707 Corn Gallons.
(.2i56,42)996,66(=o,46 CornBuftiels.
5. But fince Multipliers are the Reciprocals of Di^
^iforSj therefore Unity divided by thefe Divifors will
produce fo many Multipliers. Thus,
282 )i.ooooo(=so.oo3546 the Mult, for A. Gall.
231 )i.ooooo(=o.oo4329 the Mult, for W. Gall.
26^ )i.ooooo(ssxo.oo3722 the Mult, for C. Gall.
2 1 50,42) I .ooooo(=a:o.ooo465 the Mult, for C. Buih.
Confequently, if any Area be
r 282 "J f o 003546I the Area f A. Gall.
Divided) 231 /or mnlti-1 0.004329 I will be J W.GalU
by . . j 268.8 fplied by | 0.003722 rexprefied 1 C» Gall.
12150,42 J L000046S Jin .... LC Buih.
6i
^. >
y>+ Mf MSURATION applied
6, In order to work thefe Dimenfionis by the ir-
iijicial Line of NumbetSy either on the ffain Scaler
or Sliding-RMle\^ put B=Breadth, L=Lenjgth, D
=Depth or Thicknefs, G=Standarci Gfallqn \ and
AssArek, arid $=Solid Content in thofc dallons^
(^c. Then for Superfieies; ^ = A 5 and fo LB
» AG ; therefore G : B i: t i A. That is, the Ra-
tio^ qr Logarithm of the Ratio of th^, Standard Gat-
Ion to the Bfiadtbj iS equal to that of the i?^i//i' of
the Length to the jfr^/? in Gallons. Wherefpfe fup-
^ofing 8=31,57, L=4i,5 (as in thitPdraileiogram
r\g. XlII. Art. 3.; if you fet one Foot of the Coni-
paiic^ in 282, and extend the other to 31.57,' thajt
Extent wili reachi from the Length 41,5 to 4 t¥
Gallons, Ale Mcafure. Or, on the Sliding-Rule^
fet 282" on the Rule to 31,57 on the Slider, then
againft 41,5 on the Rule is 4 -n? on the Slider, the
jlrea in Ale Gallons^ as before j and thus the Area
is found in Wine or Corn Gallons.
7: Again, Since AD=Sx i, therefdre i : A::D:S.
That is, the Logarithm of the Ratio of Unity i &
the Area in Gallons A/ is e^iual to {hat 6f the Ratio
of the Depth (or Height) D to the Solidity (or Ca-
pacity) S, in Gallons of the fame kind. Where-
fore if 3=21,56, L:^3i,57y 0-^:9,03, as inPa-
rallelopipedon, Fig. XXV. then 0=282 :B=s2i|56
:: L=3i,57 : A=2/<y the Area in Ale Gallbns ; cori-
fequently i : A==2 t^ :: 0=9,03 : S=2i 1%, the fo-
lid Content in Ale Gallons. Where ejttend the
Compafles from Unity to the Area 2 tV; the fame
Extent will reach from the Depth 9,03 to tht folid
Content 21-1^5 Gallons. Or, on tht Jliding Rule, fet
Unity I on the Rule to the Area 2 1% on the Slidef»
then againft the Depth 9,03 on the Rule you have
21 rs the folid Content in Ale Gallons on the Slider.
Thus the Method of finding the Content or Capa^
city of fquare or i^dlilinear Arecfs at Bodits in Ale^
fVinei
' t
Ch. XV. \ ^ Gaugino; ^ 225
fTine^ dr 'Corn Gallons is exceeding jflain ahd eaf;^
by the Single, Gunter and Sliding- Rule.
' 8. In cafe oi circular Area% lince tfiey are all in
the Ratio ocf ,the'Squafes of their Diameters v and
fuppofiiig the EMameter' of ^Circle i Inch, the Area
will JDeQ,7 8^ P^drnal Partsof a Cubic Inchi. tHore-
fpre having the Diameter of any Circle given in
Inches, if the Square thereof be multiplied by. 0.7854,'
tlife Prodaft-wtlH)e the Area df that Cii^cle in Cubic
Inches. ' Let Diithe Diainetert)f a Circle, 3=0.7854,
GidStari^dard Gall, and A==Area"bf the Circle in
fuchGaflons, as^efbrc. Then i : a :: IJP'! DDa=Area
in Inches; therefore DDa=GA*5 and fo G : a :: DD
: A = Area in Gallons. But a : G :: i : — s: the
Square 'of the Diameter of that Circle, whofe Area
iOi WhBttfore fince DD = J A,
<hereforJifthcSqu?rero.7854) iSz.cooofzr 359^^5
,oftheDiameterDDjo.^854) 23i.oodo(= 294,12
be:dividedby2b=/o7854) 268.8ooo(= ^iM
. * • .(,07854)2i50,4200(=2737.9:2
or the feveral Diyifors 359^05 ; 294,1^, Cs?r. then
thcfe Quotients will be ( the . Ajrea of the. Circle in
JlC;^ fFtnCj &CC. GaKoncL
9, Or tha?,, fi»|cc DD x ^ =5 A j therefcre if
the Square of the Diameter DD
U U'VaX'-^^^ )0 7854(0,002785
be. multiplied I ^^i )o.7854(o,oo3389
, by^= i2688 )o.7854{o.oo2922 . .
... . C2 1 56,42)0.7854(0, 0P00036 "\
^jfeveral Produfts lyill be the Area in Ale, Wine,
(^c. Gallons, aS' before, Suppofe the Diameter of
^Circle D=;50 Inches, then DI>==2500 ; and put the
donflant Divifo'^ j =dd= 359,05 •, 294,12, Csfr.
G g Tlien
j^2$i Mei^v iA^lov Jif>plied
Then' fuicc ;xDD=:ddA, therefore d^tPDj: pr-
: A= the Area in Gallons. Therefore with the Cxjm-*
pafles fet oiv Foot in 359 1O5 extend the other to
a^QO, the fame Extent will reach- from i to 6 i^z^
the'^Gallpns in the Area^of the propofcd Circle.
10. Otherwife thus-, put ^ =^=o.o62}f85 j'
0.0023^9, (^c. (fee Art, 9.) then bccaufe DDdd
= A X K ; therefore- "we- have i : DV}:4idiA i or
i:dd:i DD : A. If. then jfou fet i cm the Rule ta.
0.00278 on the Slider^ you will feea^inft ^500 omt
Rule 6-^ on thp Slider^ the Gallons of Abin.the
Area of. that Circle, j; and thus you find the Gallons
of Wine, Corn, &c. both by tht Sliding- Ruk and
Thin Scale. -
11. .Again, putting H= Height. of aCylinder^
and X) = the Diameter of its circular Bafe ; allGo S
= the folid Content -or Cipacity in. Gallons 5 thcii^
dd' DD :: Ti : A ::) H : S. Now let D = 50 Inches^
gnd H=i5, the Area of fuch a Cylinder; will be
thus found by the SHiing-Rule, Stet J595O5 en the
Slider, to 2500 on the Rule ; then' againft 15 on
the Slider, is 104 oh the R:ule';,atxd fo m^y Gal-
lons of Ale would fuch an hollow Cylihdcr contain. ^
12. But* fincc^d-'r I :: DD: A,^ (Art. 9.) there-
fore d : I :: D : /A ; and for the feme Reafbn d
: /*B ::*D : V"^. Confequehtly , if a Jingle line of
Numbers be made to. Aide by a double aney if; you fet
d on tht Jingle one to i on the double oi^e^ then againft
D on the former ^ you have A on the tatter, Alfo;
if againft d on the Jingle Line^ you* fet Hon the
double one ^ then againft D on Xht farmer," is S on
the latter. By this means therefore/ having only
the Diameter and Height of a Cylinder^ the Atea m
the Safe, and folid Content of the faid'Oflittd^r/ h
immediately known. The Reafon of thS' Method
here.
Ch.XV. ' ito Gauging; ^ ^^^^
' hett tfeEyifred^ appcats from wiat has been altetcfy
iaift in the Chapter of /^ri^i?*/<y/ 0^rif//<ww. . ^
'^ V3-'^ow d =
V 1^9>05= j8,95l^yyhicli Numbersf ^^ Gall,
^$/ 294,12= 17,15 ( are called the] Wine Gall.
•: 342^24=:i8> 5f G^^^ jP(?i«/ijCprnGaH,
^1^55737^2=52,32 J for [^Com Bufli.
And accordingjy on the finjgte Lirte of Numbers a-
^ gainft thofe Gduge pMfts there is a fniaH Stroke (^},
1^ whicfr are itt the Capitate AG, WG, MB, to
figmfy tlrey are Ae Omg& Pmnts for -^/^ Gstlkns-^
JVine Gallons^ and A£j// Bufhels.
.14, Thefeforehi the^nemcncioti'd Cylinder, whofe
Height H was cb 'i5 Inches-; and the Diameter ci
i$.s Bafe D== 56^ in order to find the Area of the
Bttiev and Capacity ^ the Cylinder in Gallons, by
the Sliding- Rule ; fet i on the Slickr to the Gaugt
t^ini for Ale 18.9^5 i)(i^t\^% Jingle Line^ then againft
the JDiamiBtcr D = 50, on me fir^e Lme is 6 A on
the double Une on the Slider, the Gallonl^ of Ale in
the Area jof the Bafe.
• Secondly, Set tYi^Gauge Pm4 1 8. gs to the Height
15 ott the Slider, then againft the Diameter 50 oa
the ^)gle Line is 104=: Son the double XiUxe Or
Slider \ therefore the. Capacity of the Cylinder is
104 (^llons, Me-NUafure^ as before.
15. Or thus, with the jfe^& Line' and Compajfes
only.* Set onh Foot m thz Gauge Point 18,95, and
extend the other toiD =e 50, this Extent turn*d
twice over from i will fell on 6 A 3= A = the Area
of the Bafe in Gallons jo( Ahsi The folid Content,
in this Cafe, will fall beyond the fingle lizie, or elie
might be performed in the £im6 manner.
16. With the double Lines of Numbers, on the
Slidiiig 'Rule, thus. Set the Gauge^Point 18.95 on
^the Rule to the Diameter 50 on the Slider, and mark
the Nunibcr on the Slider againft i on the Rule •,
G g 2 then
«d«8 Memvuation i^lied ^
then agiimft that Number on the Rule you E^ve
6 ^fs == A the Area of the Bafe on the Slider. Alfb
mark the Number on the Slider agsii^ft the Height
15 on the Rule, then again^ that Number on tl^
Rule is 104 on the Slider, the Capacity of the Cjr-
lindcr as before ; and thus both the Area and Soli.-
dity are found at once fttting |hc Rule. The fatnac
is performed by the double Line and Compaflc4
thus; .Set one Foot in the Qouge P^nt 18,95, ^^
extend, the other to the Diameter 50, the (ame Ex*-
tent will reach at twice from i to 6TWthe Area^
and from the tjeigbt 15 to ;i 04 folid Capacity in
Ale Gallons, . .. , j
* j;. In this manner niijiy Gaugt-Peints be found
for right-lined Areas: for kt a^yfuch Area givea
in Cubic Incb^^ be called a y then (hy Art; 6.)\ i x^
^s GA, and fo G : 4 :: t : A ; or thus G : i :: ^ : A ;
therefore •& :!:: s/mi i/^.
Lv^2 150.42=46.36=3 LBuffi.-
Call thefe Gauge Poinds d ; then d :jj i/T: • a =
the Area in Gallons. Alfo i: v^H:: v'«:^S =
the Solidity in Gallons. Whence you. may obferve;
that rigbt'line'd Areasj as well as circular dnes^ may
be found in Galions, by the^^/tf-and douhk Line of
Numbers Aiding by each ^ other ; but not the Soli-
dity of fuch Solids, there htrag ikrie^erms of the
four variable, in the Analogy, for that,
18. But notwithftanding this, there is a Method^
whereby the Solidity or Capacity of Solids or Vefleb
may be^Aind /without knowing the Areas at all)
by the Brendth^ Lengthy and Depth, only 9 which
call B, L, D ; and S = Solidity, and G = the Gallon
or Bufiiel, as before. Now fince BDL:=:GS, thttc-
fore BD>£:gs = Gx^xS; andfoGx^:B^
Ch.XV* , to Gauqimc}.. ^z^
:: D : S. But bccaufo j;^;^ thjB RedpflDod of I^
therefore the Line of Numbers on which you feek
Xi muft he inverted i and becaufe the Logarithm of
G is added, the Number G muft always begin the
invertffd Lincj or be placed equal to i on the dire£l
Une. Thefe things premifed, 'tis plain that if B oti
the Slider be fct to L on the inverted Line ordered
as before, then againft D on the direff Une you
hatre S dn the Slider.
19. For Example ; Suppofc a Ciftern be 80 In?
ches long, 50 broad, and 40 deep* Quaere the Con-
tent in Bufiiels ?
Hert G= ^150^2, ahd for thePurpofe of Malf^
Gauging tfiere is on (6mt Rules an inverted Line of
Numbers fix'd on one Side the Slider ;. bjsginning at
2150,42 as before iaid ; then on fuch a Rule fet
50 ss B on the Slider to 80 = L on the inverted
Liiie, and againft 40 = D on the direff Line on the
Rule, you have 74=S=the Number of Bufhels the
Ciftern will c(Hitain. ' And thus you might proceed
for Gallons, had you inverted Lines beginning at
282, 231, 268;8,
20. In gauging Ca/ks^ the principal Conffdcratfon
is the Curvature of the Staves ; as A B D. Fig;
XXXIX- which G«^^r/ reduce to /wr D^^r^^j, or
Varieties^'' vizi
Variety I. Thofc {Caifcs whofe Staves are njoft
eurved or tent^ are cbnfider'd as the middle Zone or
B-ufium of a Spberoidy fuch as Fig. XXXI V.
Variety II. If the Staves are not quke fo much
crcbing or bent^ the Caik is fuppofed to be the mid^
die Zone^ .or Frufium of a Parabolic Sfindlej a$
i%.xxxyiL
Vfiriety III, When the Staves of Caflcs are. but
very little/ ^«ry^i, they are reputed to^bc^in the
Form of the Fruftums of two equal Para folic Co-'
noids^
kjcr MENSUltXtioN appUed ' ' '
waids^ joim^d together it the: wideft 6ale9. ^ jl^/A
XXXVL
V Variety IV. When tic Staves are ibait from the
f f^;/^ to die Head, as the prickVi lings Afi, BD^
for very nearly do) ihtn 'tis plain Xuch a Cafk con-
lifts of the Frtf/iums of two equal right Gones^JSst
together at the greater Bafes. Fig. lOCXUL
21. The Calks being reduced to thefc ibur Va-
rieties, if you multiply the Di^erence between the
Head and Bung Diameters
for the < ii/y^wctyy
and then add the Produft to the Head Diam^er^
then that Sum (hall be a vman Diameter^ or thajt c£
a Cylinder^ whofe Height -and Capacity is 6(|ual
to that of the Cajk^ as near as poffible.
az. For Example •, Let there be a CaOt ADEG^
whofe Bung-Diameter BF = 3 1 .5 Inches, and Head^
XHameter AGr=24,5i then their Difference is BF
— AGp:;.
r r .1 (rJI^'^Jll^yThe mean Diameter
Confequently\7xo.6=28,7 / ^f .^e Cylinder
24,5+ ]7xo.65=29>05r equal tpthic#.
17x0.7 =29.4 3 ^ "^
Having found the Areas (by Art. 8,9, 10, i2^)>t*
be 2,2385 ; 2,2941, ^c. Ale Gallons j andfuppo-
fing the Length of th^ Qiik 42 IncheS, ssHL.
Then the Contents f2,23«5x 42= 94.0^3!
according to the 3 2,29^1x4^= 9^-35l Ale
feveral Varieties 7 2^3 504x42= 98.7x1 Gallons,
will be : ; . . ,.{2.4073x42=101.103
And
Ch.XV. /fl T;MBER,-^KJiwm«: 131
And thefe .being all the princtpal Articles, w^heron
the yfe of ififtrumental Logarithms^ or tk^ artificial
Sne af JsTumbers is diredlr conoerned, I fliaU fay
m^ on^hc Head of Qau^ingy \^^ ]pr^^ed ta
thp nejtt Articje of ....
r-.:)
X^mier^iJ^^furc].
2 J. Ev^ry Piece of ^mber is a Solid, l^kcto fiunje
one ijf other of fhofe, whofe aifoluti Mensuration
was ftewn in the JaU Chapter, viz. The Frujium of
zCdnt^'VtyUnder^ tilt Prujium qf a Jpyramid^ zPa^
rknelopipedon', Prf/mj 8ci;.[ , -.,"?..'.
' !24i In .w||iatever Form the Piece h^pens to be*
find the aintint ox Solidity Jn I^hes, ^ there taught^ s
tK^hich'divide by 1728, (the /olid Inches in on^ Foci
dp Tinjibcr) the Sluodent is the/olid C^tftent 14 Feet^
But ^H^t^ the Analogies for Operation, by Inftni-
jTiients^^^ky be evident,^ I ftall m^ ufe of the fore-
going Method in Gauging^ by putting B = the
Brcadfit,' D=pepth, t.== Length, G=i2, and
S =*:^qiidity, aUiiii \m\9k. Then" fmcc ^= S,
therefore J5I>L=:;G^Si but J^ = thic LrngthvoiFcet^
^hich.lct be F, then BDJi=GGS = BDF ^ and
confequently GG : BD :: F : S.
25. No* if the Piece of Timber be in form
of a fquare Prifm^ then t£e Bafe BD' is % buar&
NMmber^ which dall gg\ whence CGzfigr.¥:S \
and thefcf<we.G:f ::v^'F:v^S. Wliercfore, bar-
ing ^Jingle afiid double IJjh of Nmbers^ by the Sli^
ding'RuJe^ fct G=: 12 on the^ Jngle Line^ to th«
Length in Feet F on the double one; then againft
g on thp Jingle Line is S = Solidity on the double
Line oh the Slider. Example, Suppofe a Piece of
Timber, the Side oJF whofe fquar« Bafe is Fm^
inches.
ft§2 MENStrRATTio^r bppHed ' '
Inches, the Length i8 F^±: F; Quere thi folidl
Contents? • ^ ....:. r
: Set the dbnftant Point G = 12 ob thc/nghZine to
the Length m Feet F r= x8 ; thenagainft the Side of
the given fquare Safe |^ = 15 Inches, on the former,
is 28 = S = the Number of the .folid Feet in the
Piece, on the latter. Or with the Compafles, ox-
tend from. 12 to 15 on ihc^ugle Line^ the iame will
reach from i% to 2i8 =:^.S in the dpubjc Line, dje
.26. But If you have no//i^/^ tine^ proceed with
the double Line on the Sliding-Ruk uimx Set the
conftant Number GG= i^ on the Ruje.to, the giveri
Squ^e ^==2 2 5 on the Slider ; then 'agaihft the LengA
inFeet Fr=:i8, oji the Ruleis^ =$ = tHe folS
Feet on |the Slider. ^Vy with the Cqmpades^. ex-
tend ifoia i44r ^0*225, the fame Extent^will jcacii
fronj' 1 8 to 2 8 the Tolid Feet, as before. ' Otherwiie
by the Analogy G t ^ :: ifF : v' S ; thus, fet thexon-.
ftant Point G= 12, on thq Rule to the.- given Sidp
^=15, oh the Slider, arid ^ mark 'the Point on the
Slider againft 15 cntheRutei bi-ing that Pbint t6
12 on the Rule ; then jig^r^ft F:^ iS.onJthcRule^
id 28 tzS oh the Slidevy the 5ip//W//y as before. But
much better with the Cqinjjsifles,. thus ; J^xtepd from
12 to 15, that Extent tui^iiM fwiec ovtt will teach
from 18 to 28 =S, as befotc. ' : - ' ' '
• 27. fif thie Keccll^e ife the Form of* a Parallelopu
ped&Hy t^at' is, hath its Breadth and D^pth uneqoal^
then the common : way vis ;tp: atdd iht .Depth and
Breadth tQ^tthtt\ and to t^ke half thatuSvnn fen: the
Side o^ a mem Square; */». ^^^ tssg^ and then
they imagine ^^ = BD, andfo mcafure ^ihe Plege as
before. But this is a very ' erroneous way v^ and the
more fO| as the Difference between the Breadikh and
pepth is greater. For fince, (in order to reduce thi^
Piece to a 'fquare Prifm) BD = gg\ 'tis evident
fch. XV. to Timb^r-Me/A^WJi^: >a J3
|rte£s i/BD i that is, 'd^Anean.Prtfporthnul betwefen B
:^d D, (for B: VBD:: •BD: t)) knAyi^^xk^fbilf
^ ^."«" J as they ighorantly; fuppoftl''' .\
28. For Example ; Suppofc a piece ofbewn Tim^
#rr • iaform as aroreiaid, 'whofi Bicadthis^a^J In-
^-^hcs=±: B, the^ Depiif or 'Tbickaefi-to Itches as !>«
* imd Lcn^h 18 Fceir ifc T j whit is th« f(jlid Con-
.j: .2.9. Find ft wf/?/r Prpfsr^ansi bet^Vceti; 8=^2,5 ajid
^ b = 10, (as heretofore taught) it (hall i)e ^. =± 15
'i==V5S*i' and thierefore 6**= 144 :^*'22|l: Fd:i|8
: S ='2S(,*jas»befofe. ' But accordiij'g .to the/comrti6a
Vhich^s greattfr'tKah- the true Area^of the^Bafe BD
<^'2r5 h^i^^x;>6i^fquare Iftehs/^ AKb f* Gzr I2
•' 6iiith«>fwg^ LOU of ^^Numbers t6 theJjBngtfc F set 8
'^f*ect «fa ttte dtMbl^:lhen^^auiff^Ikiilfe)^==j6,25
<rP^i?be^/irwi^, is .32! a? S oathe ktt'er. _But 32!
^— 2S = 4! Feet more than is reaQj J!,n 'theTiece^'
-• ^"'igV-^Becaufe' :B*I>'F te G^S ;'- therefore BD =
v^ Si \tid fo.wcihacve Q\k y.RiipTS., U .then
you have 2Ln Inverted Line of Numbers begmnifig
^'fK)i»^G*fc=: I4I pHacfed oh onfc (ide^lfcerSRder^.^with a
-'j^irrlfXi^e bn the^bthir ;^ then mAy the filid Content
^^n'Feet tie fotitid tjy the®, D,' andi^i as taughr^in
Art;!?; hereof,- thus ; Set B «afti22,5 on theSHdir,
• 'to^FsiiiiS; oif *he in*uiited^IJms then againft D
rtei xo^: 00 thfidireii'JJnt k iS^^n S, on tm SUder,
. theLfpKd Feetfas before, ^
. . . SO, if ,th« Timber be In JPorm oJF the Frujium of
. a Square Pyramid^ a^ Fiff . XXX. the Theorem for ^
» it^ Sfrfidity in inches is HD^ 4" iH^-x, as pir laft
; Chfips^.A^rt, 23.; Thercfore^HDd-f iHx?c = GfS
(where S== Soliidity in c»^zV7^tff/)J and fince i'iH==F
te the Lepgth in Feet,' therefore FE)d + ifxx =
' H h G
vSdp^ a f insfi df DHmbetaf Incbid (quare bt fJ»e
gretteft end, 9 Inches (quare a( th^ lefEst Enid» .aod
ao Feet l<Mig, how many (olid Feet is f Here in Tiich
•^Tftef. . • ■• ;
I! IieteF=£2o» Dd=s2\5,9ndi)ffi^=f8-5ti^.ihak-
/fore D«I*HxK=gtOi,3"> wherefort G*asM.4dtJM»o
• v^yio^^i : 4},i=sS, the t^bmher o£ foUd F«ec ^tir-
quired. Note, the common way, by (uppofiigidie
: Sqflafe of ^ W m^-i** is very ^fej iiA in
<. this .Indance would not give* the Solidity alibve 40jf
! Feet, which is threcTeet lels than the Ttuth._
• "31. If thiyi!Stsot{lAtPruM»i^'P^'^^lifH^^^
- Vs is^ the Cale 6f moft Fi^S^f i5«eV^/«AEr^;^Uiin
• j^ey «ay be: reduced to Fniftun^.of yj«arf. i'jrw-
.. mds^ thus 4 4-et,A:ss tije Area qf ^ g^tqr'Bai^,
■' ft = Afflka o f : the l afler. i TAen, t< AaP*. «ti^jt<5a
■ s-z/i aAdVAxTasn<^i *l fo:fince D**4- a B)i
' 4. dd':=iiuc,y thetefore A^'2 y/Aka"4iai=ia<*»'iriaro
r|Saif£ ^ '4i4|S3ii "oonfeqwntly, G* : F ::
':.|!A V v'^A^H^-: $ te'the Nimbcr- of y&ir*? ^/
';-ih.fucb'a Ti5w., ,1 .;, -,. ••-'■•; . . ..:!•/•
;: v,yL. F<m: Example j Sqpppfe:* Piecft_pf /qiBJrtd
.fBxs&es'.h&jit.Whes broad^^^d'ap dwipr^vtte
uJargeft End j -aftd.; i cl bf«>a4 a*?^ 6 6e&^at theil#r
;Endi thcLei^th 18 Feeli. Qj??^ f^<t w4^otent?
:: :Bpr e:A== 3? x z Q ae ^40 » and » t=. ip xjisPi^ J
t- and" v^ Ax'a,6aV c^8400ateVi9<^959 » . there&re
■J A + f Arira4- a =55 298,653. ^TherfG''''^1^44
•:F= 18:: 2:9S;6^3; 3 = 37,^3 the folid 'Fftt re-'
pfad;
the common or cu
Ch. XV. to Tjmber-Mb A8iPfR«, 2 J| -
33. If the Timber be in form of a Cyfliodef, F^^.
XXVIII. tihcti ^tting a = 0:7854, and G =p la r
Inches, we have (as per Art. 8.) thfe Analpgy j/
i: a ::DD: DDa=the Area of the Cylindw^sfia^-iri
Inches (iuppofing Dss Diameter in Inched i) Ic^ .
Hiss Height ia Inches-, then HPDa =t the. Inches
fo'lid, whereof 1728 se G» «: one /i//<i i^<w^ ,. Con-' .
feqirtntiy, if S?s;folid »£«»/«»;( lr» Feet, BTsHDaj^:
Ci*Si and S =aF=-,tJie Length in^ Feet,; there. -
foreFDDa=5G*S j and FDD = ^ &i«n<i Pitting
^st dd « 1 83,34 i then FDD = ddS » wherefore
(i^ :'Dt) : : F : S, or d i D :: /P = *^ ' . ^^^
.. «4. Or with the Length' in Inches, to find tftfc,
Stjpty in ]Feet, thus \ fince HpDj = G'5,. there-'
fore putting ~ «^<^. weiballhaveHbD==^Si
' cftnfequently 5i: DD' ::H': S ^Solidity iii Feet j
or </ • D ■• yT5 ; *^*S. , Wherefore, if there be a
fe/ijV of Numbers on your SHdinglltjlei ^ fetthe
c&j^ht 'Numbe? d = 46.9 o« t^e^fingk |-inc..to:
^ « the Height in Inches.; on the douUe Lineoii
the Slider; then againft D « thi^ Dmnetsrrof the.
Cvlinder»8 Bafe'in Inches ont\ittJngliU»«, is S :^;
the Number of folid Feet on the^Ud«r'afr requj^d,
Aftd if 'the Length be given p Fjet j then the bpr
lidity ia found, as above. Art 13. .
-35. 'If you have ho fmSrle Line on yont Rufe, yoy
rouf work with the Anabg4«MiRP:;F:;S i or
dd iDDiiHi S. In the firft dd = 183,34, and dd
,« 2200,154 i theiefbredsFb 13.54 = Diameter of
i Circle, w&en the Area is i44:i «»»<! d =?= 4«.9 =*
AeDiameteiSofaCitcle, whofeAreais 1728 ^fthq
Inches in a folid Foot. Andltfctfc Numbcts, bemg;
cSh'flint itt all Opeikion*, and the I^Icthod of opei
nitinS Cl^elvfcfeA'pt Analogies ttcry way o« the f r
■t-.-F'^ Hha 'y*""^
tijicial Linis ofLogaritbmi already, fiiffideotly exam*
plifkd in ^ proceeding Aftides ; I (hall noffaerp^
again re|)eat it.
3d. The cominon way icf meafiiring round Tim^
beTj is l)y girting * them about the Middle with a -
Siring^ and taking lof the Girt for the Side, of z
Squats equal to a vtean circular Area^ fach as would
r^uce the propofed Piece tx> aXj^/f»&r^ But this
isLalfo very mfeat^d-ungeometricaL' For; (in^e the
Area of that Circle, whofe Circumfefen^e is" i, is
o.a795^ ; and-thc- Area of that Square, whofe Side
is ,25 (=1 of the faid Circumference) is 0.0625,
and the Solidftics bemg Ml proportion to thefe Num-
bers, W2. as 0,07958: 0,0625 ; that is, as 2 3 to 18;.
'tis evident,, the Content by this falfe way is above
4 lefs than what it really' is 5 which Err or ^ if it be
not confiderable enough to be regarded and correfted,
is great pity indeed. i
; yf.. Therefore to-mea&re.a-Picca of rmnd tape-,
ring Timber truly, it muft^be confiUered as the Fruftum '
of ji rigbi. Conei vyh ofe. folid 'C ontent in Inches is
found by the Tlifiorem 0^4-1x^x0.785411^ as/<r •
Art. 25. of the laft Chapter. Now putting 07854.
*tt'a,.and G' = 171B, the cubiclncnes * in a folitl'
fbct; the Theorem will become Di 4";}*"^ xaH
=: G^S ; and ag ain (ince i\ H =5 F, the Length in
Feet ; therefore Vd + 4^^^ x a F tsG^S i alfo put
$^ sss^d; and then 'Dd-^^ixxxF =zdiiSy where-
^re we have dd I'Ua^yxx :: F : S =?? th^ folid Con^.:
tent in'Feet.'
■ 38, For Example; Suppofe a Piece of. round.
Timber be 36 Inches Diameter at one £od, and 9
Inclies Diameter at thei other ; and 24 Feet long j
cuasre the iSolidity in Feet ? .^ i ,
^Heit 0=^:36; i/x=9, D— .i/wy=:.2'7, Prf«.
324, ^*?«=ki'45i-I^ + "^^«567» Wd? = 24.
Ch.XV. . /^ Measuring ;Lanp* -^ 237.
Therefore the Anelogjr is as dd=: 1 83^34 : 56;? :? P
=?; 24 : S :;='74,'22 the foljd Feet reqi^ircd ; a3 mayj
be wrought by the Lines of Numbers in luiy of tho;
bcfore-mentiotfd ways. * • *.- ' ' z '
0/* Measuring Land; >• :
39. What Iprineipally defign here, is to fhew^
bow the true Area or Contetit of a IMoT'of Lftnd is
to be found by the arterial Line of Numbers^ in
^cres and Decimal Parts. The Dimenfions 6f a Field-
are commonlv taken in Rods or Poles i each con-
taining 16 f Feet. Of thefe Poles, 40 in Length
and 4 in Breadth nuke an Acre ; or an Acre is zsz
i6q fquare Poks. Some (and indeed moft) ufe ^
Chain, called Gunter^s Chain^ in taking Dimenfions^
which conHfteth of 100 Links, and the whole in
Length = 4 Poles or Rods. And fo 10 of thefe
fquare Chains make an Acre.
4Q. The Field being meafur'd with the Pole, if
it be in Form of a Parallelogram^ put L = Length,
and B z=: Breadth ; and then it will be — s= A =
'- 160
the Number of Acre$ ; therefore LB = 160 A 5 and
fo we have 160 : L :i B : A = the Acres. For Ex-
ample» fuppofe a Field be 35 Pole broad, and 185
Pole in length 5 how many Acres doth it contain ?
Set 160 on the Rule to 185 on the Slider, then a-
gainft 35 on the Rule is 40! nearly, the Number of
Acres required.
41 . If the Field be in Form of a Triangle, and
meafur'd with a Rod or Pole ; then, having plotted
it and .meafur'd the Bafe and perpendicular Height,
which call B and H ; then B x f H = 160 A j and fo
160 : B :: IH : A :;= the* Acres as before.
42. If the Field be in Form of a Trapezium^ as
Bg. XL. then becaufe, by drawing the Diagonal
AB, it is reduced to two Triangles ACB, ADB,
and
3538 • MENSCRATrON^^/Sflft/, &C.
and ealfing the two Pcrpcndkrulars Cr and W,* H
«&d ii and the commdn Bafe AB^JB^ we have
^H-^»H+TxlB = i6pA. Therefoii 169.
) IB :: H 4- ^ : A =? the Acres contained in the .Tra-
pezium ACBD.
43. If the Field be of a mult angular Farm^ it nHift*
when piottedt htt rediKed tp feveral ^rianghs apd
trapnia% VfA then meaAir\i, as ;)rr Art. 4^4^-
In csife you take the Meafiires with a Qhaixxof 4 Rods
ilr I to Links^ then^ the Anabgies for Op^^tioA
will be the fame as above» only inflead of 160 you
miift iufe . 10 \ thus IQ ! L :: B 3 A. Art. 40* and la
::B :: fH 5 A. Art. 41, fcfr. All which is fo ^fy ^«
to nerd no Example, nor any thiag more tQ b^ ^vA.
COdcemit^ i^^
^ C £[ A Pw
, ^G HAP. XVL. , , :-
other refpcfls it is very little. Bati'ih feding^tie
Places, Diftances, ^c.-o^th^ h^venly^BoSes^hs
Calculation of JEclipfes, t^c. tljey are Ter^^'.neco^^
"f6r'lfiiidihg the f rcyoi-tibnal Pstt-ts/-as;^tin,^ in fomc
Degree, appeaf i)y the 'following Eramples. -
' a.,\ E]f|b}^j}te :i ; A^mk,: the meun Idftoifm^, ^f Athe
-^aad-JLogarithm of hiiJ>i&ince fron>4h<hEarth~?
I' Mean A- c 4* f )Eq!aa^.i° J3'49^a^
nomaliest4 8 J tions C i 3236 J^ar.C 4.99.5501
DifFerences i=$d'
Thci^ fbr :tkt:TrbportmalJfarts.o£.tht EquB^oaii fcy
by thc-tiOgiftical Logarithms, ^
• '/giye the jQiffisrence-, i 13 =;i()930
. wMt«iye4rthe^?/^0pmly ,., . ,9r4fir?J*fe. ^^8
Anfwer, thie prop. Part ..... o 59 =1^7878
This futyftfe^ed, (b^caufe the ^fqu^tion is- dc*
^^creafing) from the Equation i^ 32' 49^ (anfwcring
I tht
\,tt4o • f35^ PftAtTifcAL Use ^JfiW
the Anomaly 4* 7°) leaves the true Equation :S*
J® 32' 50^ as^was require.
-** 3. Then for the' proportional Part of the Lbga-
rithm (ay
One Degree, oi»^ . ; % . .-. . / ; J ; . ;^ 60' 00^= o
givy t^^ Diffcrcnccx>f lx>gu-ithms . ^ io65?i53ia>
what gives the Anomaly . . . *. ^ .. . 148' 1 4^= 948I
Anfwer^ the Proportional Part, i * . . * . ^5=16258
^VhichTubdtt^ froni theXc^rithm 4,95560^, ah-
. iwering-to the Anoip. 4* 7^. itere.remain54,95B522,
' the true 'Logarithfjf of. the ^un*s DiJianc,fttom the
. Eafth^ as requii^fd. :
^. fixkihple. IL, Sdppbfc ' the Moon's anhual Af'
^g^^^»/ bj5 29^ 51' 37*, what is thelEquiattion of the
jipogee inSi ^tcmtrhUy Qi}itv Oxh^ ." \.
- ' Amiual t^^ I Etfus*- J 8*^'$3^ &^ 1 Ecc«i- 1 6^392
. Argum;t 30 5 tteii (9' 07 i4 i ttiikyV6to4s
v.-' j ( .> v -> ) ::i ii i G Gill r .*^ .: > ■! ? ' *
^Differences . i=;.6o', 14 6 ^ , 65i
T^ ' . ^ ;^ r. ■ ^ i / 'iiii .1^ ii "] . } ' ' 1.1* III I I
- Tlifcn for the JEquatbn of the Apogee^ 6y
lif pUfiJOegree, ot* .^ ♦. . , ^ . : . . ^o' oq1= ' 6 ' '
give the Difference . • . * . . 14' 6^^=6289 •
V wlwtJgiyqi:! .ri.. * . * * ^:y^V''^^t^ ^'^j^is: .6^4;' '^
Anfwer, the Prop. Part • • * . I2' 8^ 6943
*^T9 Which^dti" the Equation 8^ 53' ' 8^,' kgrfeeirfg to
<r:: ^ ' -^^ II I ■ * ' . , - "li thtaft* Arg*
<^tt^^ j^hE'^uii^is . * w ..Vg"^ 6s^ 16^ ths trm E-
ijrisf27/rOTi of the Apogee as required..
5, Thc;i for- the true Eccentricity, fiy ,»
*. 1 1
Ch. XVI. Logistical Logarithms. ^41
If one Degree, or . , 60' 00^= 9
. give the Difference of Eccentricity . . 653 =7414
what will the Part .,...•• 51' 37 = 654
Anfwer, the Proportional Part 562 =8068
To which add the Eccentricity . . • 60392
The Sum is the true Eccentricity . . 60954, as re-
quired.
6. In like manner the Equation of the Moon*?
Center, Latitude, Inclination of Limit, and Loga-
rithmic Diftance from the Earth may be found. Alio
the fame things are in like manner foimd for any
of the Planets, of whigh there needs no more Ex-
amples.
7. Ejcample III. In an EclipTe of the Moon, ad-
,mit her horary Motion be 30' 31'^, and the Sun'^
2^ 27^ ; then the hbrary Motion of the Moon froni
the Sun \^ill be 28' 4!^ : and fuppofe the Moon hath
pafs'd the Sun by the Diftance 1^ 19' 4'''=4744'^i
What is the Time requifite for that Motion ? Say
thus.
As the horary Motion q( the Moon C • ni
fromtheSun { *^ 4*3300
is to one Hour ; 60 0= o
So is the Diftance pafs'd above 56' c , g_.
&\ viz. . 1 ^ ^^7'
-*»
To the Time above 2 Hours, viz. . . 49' 00= 877
Therefore 2^ 49' were pafs*d fince the /r«^ P/>p^/w«,
gr Moment of the Eclipfe.
Note, becaufe in this Cafe the Motion of the
Moon from the Sun, performed ip i Hour, is 28'
j^^ ; therefore p,6^ 8'' will be pafled in 2 Hours : But
^e prefent I^iftance of the Moon frpm the Sun is
• \\ 79[
Z4:Z ^^ Practical Use oftbe
79' 4^, which bccaufe it exceeds the Table of J>r
giftical Logarithms^ therefore fubduft the Motion of
two Hours, viz. 79' 4^ — 56' 8^=22' 56^ 5 and this
Excefs. of Motion will give the Excefs in Time a-
bove 2 Hours ; as in the Example. And thus you
proceed when the Diftance of the Moon from the
Sun exceeds one Degree, or the Table.
8. Ejcample IV. Suppofc, in a Lunar Ecliffey
xht Semidiameter of the Moon be 15' 15^, and the
difference between the Mootfs Latitude^ and Sum of
the Semidiameters of the Moon and the Earth*s
Shadow be 9' 01^' ; Quaere the Digits eclipfed ?
Note^ the Semidiameter of the Moon is always
fuppofed to be 6 Digits^ or equal to 6 Degrees^
Therefore fay.
As the Semidiameter of the Moon 15' 15^= 5949
Is to jSx pigits (P 00' oo'^'zuioooq
So is the faid Difference , . . • . 9' i^'= 8231
To the Digits eiclips-d 3^ 32' s^"z::,\^^%^
9. Example V. In a Lunar Eclipfe^ let the Scru-
ples of Incidence be 30/ 17^^, and the horary Motion
of the Moon from the Siin be 28' 47^ j to find the
Time of half Duration ^
Here becaufe 28' 47'': 60':: 30' 17^: a fourth
Number greater than 60', and fo confequently be-
yond the Extent of the X^ible; therefore (as in
Art. 7.) from 30^ ij" fubduft the Motion for an
Hour, viz. 28^47^, and to the Remainder i' 30^
|ind the Tiinc thus ; ,
As the Motion . , 28' 47^= 3190
Is to an Hour ' ... 60' 00^'= o
So the remaining Scruples i' 30^=: 160 21
To the Time 3' 8^=1283^
^ ' *"' The
Ifch. XVI. LoGISlriCAL L0GA«lTHAiS. 24J
The Tij^ne ibught therefore i^ 3' 8^; the doubte
bf which is 2^" 6' 16^, the Time of the whole Du-
ration of the Eclipfe*
10. Example VI. If the diurnal Motion of the
JSun be 59' 8% 'what is the Motion for 7 Hours
Say, as one Day ...... .24^ 00' ob^rr o
Is to its Motion ........ 59 8^= 63
So is the Time • . . 7** 15' 0^=5197
To the Motion therein • . . i if $2^=t526o
. II. Example VII. If the mean Motion of the
Moon in one Hour he 32' 5&\ how far doth fh«
move in 17'' 45'?
Say, as one Hour . . i* 00' 00^= 13 802.
Is to Its Motion .... 32' 56^=^:20392 ^ .
So is * * . . 17'' 45' 00= 1309 l^d«
\ 21701
To- the Motion required .,9^ 44' 00^^= 7899
Thus the Motion in 24 Hours, or one natural Day,
will be found 13° 10' 35".
12, Th^ Logiftical Logarithms may alfo be ufed
with common Logarithms. Thus fuppofe you would
find the Logarithmic; Sine of 18^16^47'', proceed
thus > ' '*
The DifFerences . . i'=6o" 383
\ • i'i 2 ■ ' . Novf
244 ^^^ Practical Use of the
Now the Proportion is, 60' : 383 :: 47 : the Pro-
pottional Part. But fincc the Logifiical Logarithms
of 47^ and 60^ are reciprocally as thofe Numbers,
and that of 60 is nothing ; therefore (the Differences
of the common Logarithms being proportional to the
Numbers 60^ and 47^ fay.
As the Logiftical Logarithm of . . • .. 43^= 1447
is to the Logiftical Logarithm of . . . 60 = o
So is the common Logarithm of . . . 383 =2.5998
To the common Logarithm of the 1 q « ^ r - »
proportional Part I ^^5 =2.45d«
Which add to the common} ^
Logarithm . ; } 9.496154
The Sum is the Logarithm • . .9.496439 of 18^ 16'
47'^ as required. And thusjproceed in any other Cafe
of like nature. ' ^
13. The Logiftical Logarithms may be ufed like-
wife with the Logarithmic Sines and TangentSj as in
finding the Paralkxes of the Planets, &?r. Thus
fuppofe the Horizontal Parallax of the Moon 55'
1 2" i the /ingle of her Ori with the Horizon 22^4';
and her Longitude in her Orb from the Horizon
81^ 27', to find her Parallax in Longitude.
From the Logiftical Loga- 7 o - -/ ^^n_ .g.
rithm of the.Hor. Paral. r^ ^^ ^^ - 3^^
rSiae of theV ^ ^ ^ ^\-^.q
Subduft the) Angle . . . T^ "^ "" = ^'^^^^
Sum of the j Co-Sine ofi^, ^^ ^ ^,*,r»^
[Lo«gitvde..r' ^7 0=9^
viz. 18.7470
ThercrcmainstT.cLogif.Lo-} o ,i ^n^ ^^^^
gar. of the Par. in Long. ..I ..? a y
14.
Ch. XVL Logistical Logarithms* 245
14. Becaufe when the Angle of the Moon's Orb
with the Horizon is greateft 67° 14' 20^', and (hein
the Horizon, her greateft Parallax in Longitude
will be then ; and will be found as in the laft Ex-
ample, thus;
■ I ' MM
The Sine of . . 67° 14' 20"= 9.9648
Radius 90° o o =10.0000
^IfL^g'^f!"!^;^"'}""" 5«' 37"= .8o3«
15. To find the Moon's Parallax in Latitude.
Subduft the Co-Sine of the j
Angle of her Orb with > 22^ 4' o^'zsp.pS^p
. the Horizon 3
there remains the Logiftical 1 ^^q ^,/ // ir
Logarithm of .... . \ ^ 5^' 9"= 693
which is the Parallax of Latitude fought.
16. To find the greateft and leaft Parallaxes in
Latitude.
'ji^htSb'ssrr }9° 4.' 4o"=,.„37
Logiftical Logarithm of thei o ^^i 21//- , -,,8-
greateft Parallax V °° 3^ -^77^:,
Agwn,
246 fbe 1»RACTICAL UsfE, ^C
Agwn, from the L. Lmt. of 1 / ^«//— a -
Subftraft the Co-Sine of the i
grcatcft Angle of Orb with > 67^ 14' 20^^=9. sSy 6
the Horizon ^ '
Logiftical Logarithm of thee o .,/ j^// ^.^^
leqfi Parallax 1 ^ ^^ *^ ^^^^
17. By thefe Inftances the Reader will be ap-
jpriz*d of the great Ufefalneffe and Expediency of
Logiftical Logarithms in his Aftronomical Calcula"
iions \ and as the common Logarithms are laid on a
Rule^ fo likewife are thefe; and fuch a Sliding-
Rule of Logiftical Logarithms may be very ufeful
to thofe who defire to be more expeditious than exaS
in their Calculations.
pOOKS jufi BubKJhti^ frinted for J. Hodoes a$
the Looking-Glafs on London-Bridge.
Neatly printed in 4 Vol. 1 zmo. Price bound in Calf 1 2 /. embel"
lijhed <witb near an hundred Copper- P late f^ curiouffy engra^v'd hy
the beft Mafters ; tranjlated from the French by John Kelly Efqi
^the Inner Temple; D. Bellamy, of St. John's College, Ox-
tbrd ; and J. Sparrow, Surgeon and Mathematician :
I . IWT At ureDelineatbo, being Philofophical Convcrfa-
J^^ tionsy wherein the wonderful Works of Providence, in tho
Ainixiial, vegetable, and mineral Creation are laid open, the folaf
and planetary Syttem, and whatever is curious in the Mathema-
ticks explained. The whole being a compleat Courfe of natural
dnd experimental Philofophy, calculated for the Inftrudion of
Youth, in order to prepare tliem for an early Knowledge of Na«
rural Hiftory, and create in their Minds an exalted Idea of the
Wifdom of the Great Creator. Written by way pf Dialogue to
render the Conception more familiar and eafy : with a partiadar ^
Table of Contents to each Volume.
Keatly printed in OSa^vo^ Price hound in Calf fix ShilUngs {founded
en a Plan of the late Mr, Addifon) imhtllip^d tvith a large.
Variety of curious CutSj drawn and engraven by the bejt Mafi^rst
the fecond EditioUj loith *vefy large JiJditions, of
II. A Philofophical Accountof the Works of Natdre: oontaining^
I , The feveral Gradations remarkable in the mineral, vegetable'^ ,
and animal Parts of the Creation ; tencfine ta thr Compofition
of a Scale of Life, 2. A Reprefentation of the prefent State of
Gardening throughout Europe in general, and Great Britain in
particular. 3. New Experiments relating to the Improvement
of barren Grounds, Timber-trees, Fruit-Trces, Vines, Sallads,
Pulfe, and all Kinds of Grain. 4. Obfervations on the*Huf-
bandry of Flanders^ in fowing Flax; whereby Land may be ad-
vanced Cent, per Cent. By R. Bradley, F. R, S, Prqfcffor of Bo-
tany in the Univerfity of Cambridge,
dedicated and preftnted to the Honourable Society if the Inner-
Temple, {Price bound Two Shillings and Six-pence) *
III. A new Treatife of Hufbandry, Gardening, and other curiont
Matters relating to Country Afiairs: Conuiningaplainand pra£tical>
Method of improving all Sorts of meadcMV, paliufe, and arable
Land, if^c. and making them produce giseater Crops of all kinds,
and at much lefs than the prefent Expence ; under the following
Heads; 1. Of Wheat, Rye, Oats, Bailey, Peafe, Beans, and
all other Sorts of Grain. 2. Turnios) Carrots, Buckwh^t»
Clover, Hemp, Rape, flax, and Colefced, ^c. 5. Weld or
Would, Woad or Wade, Madder, Safiron, ^c. 4.- Meadow*
Padure Grounds, and the difier^t Manner of feeding Cattle,
and making other Improvements^, agreeable to the Soil of the
feveral Counties in Great Britain. 5. Hops, Foreft and Fruit
Trees, Vine and Garden Plants of all Sorts. 6. All Kinds of
Flowers, Shrubs in general, and Green Houfe Plants. 7. A
* ' curious
1
taioui Scheme of a Farm, the amiiial Expence of it^ and ita /
Produce. With many new, ufeful, and curious Improvements,
never before publifhed. The whole founded upon many Years '*
Experience. By Samuel Trcnvel^ Gent. To which are addedi,
leveral Letters to Mr. TJbomas Lintein^s^ concerning his Compoiind
Manure for Land, with fome Pradical Obfervations thereon.
IV. The Gentleman's and Builder's Repofitory ; or Architec-
ture DifplayM : containing the moft ufefnl and requifite Pro-
blems in Geometry. As alfo, the moft eafy, expeditious, and
correA Methods for attaining the Knowledge of the Five Orders
of Architefture, by equal Parts, and fewer Diviiions, than any
thing hitherto published, l^c. The whole embellifhed with a-
l>ove fourfcore Copper-Plates in Quarto. The Defigns regulated
and drawn by E. Hdpfuu and cngrav'd by B, Cole. The fecond
Edition, with large Additions, and a new Frontifpiece, curioufly
cngrav'd, reprefendng the Maniion-Houfe of the Lord-Mayor of
the City of London. Quarta Price Bound ioj.
V. The Builder'^ Guide, and Gentleman and Trader's Affif-
tant : Or a univerfal Magazine of Tables, wherein is contain'd
. greater Variety than in any other jBook of its kind; with feveral
new and uTefnl Tables, never before publi(hed ; which renders it
the inoil general, compleat, and univerfal Companion for daily
Ufe extant ; and highly neceilary for all Gentlemen, Builders.
£urve)rOr» of Buildings, Timber Meafurers, ^c. By William
Stdnuu, Price 3 /.
VI. Hh^ Country Builder -s EfUmator : or the Archited*!;
Companion for eftimating of New Buildings, and repairing of
Old, in a concife and eafy Method^ entirely new, and ^i ule to
Gentlemen and their Stewards, Mafter Workmen, Artificers,
or any Perfon that undertakes, or lets out Work, ^y Wilfiam
Salmon, The fecond Edition, revifed and correded, with large
Additions, by E, Hoffpus, Surveyor. Price bdund is, 6d.
V\l. A CoUeftion of Novels and Tales of the Fairies. Written
by that celebrated Wit of France, the Countefs d*Ams, The
Third Edition. Tranflated from the beft Edition of the original
French by feveral Hands. In three neat Pocket Volumes, 1 2mo.
Price ys, 6d,
Seauii/ulfy printed on a fine Genoa Paper ^ and new LeHer,
VIII. A compleat Hiflory of -the Empire of China^ being the
Obfervations of above ten Ycalrs Travels through that Country: '
^maining Memoirs and Remarks, Geographical, Hilloricai^
Topograj^ical, Phyiical, Natural, Agronomical, Mechanical^
Military, Mercantile, Political, and Ecclefiailical ; particularly
upon tlieir fotteiy, and varnifliing Silk, and other Manufa^ures^
Pearl-iiihing, the Hiilory of Plants and Animals, with a Defcnp- .
lion of their Cities and Publick Works, Number of People, Man-
ners, Language, and Cuftoms, Coin and Commerce, their Ha-
bits, Oecohomy, and Government, the Philofophy of the famous
\Ccnfucius, With an Account of the Conqueft of China by thjB
^artarsy and many other curious Paxticulars. Written by tjie
learned Lenjois le Comte, Jefuit. A new Tranflation from the bcft
Farts Edition, and adorn'd with Copper -Plates. Price 6 /,
LOGARITHMOLOGT.
PART IL
CONTAINING
A Threefold TABLE or CANON
O F
LOGARITHMS
I. A Canon of LOGARITHMS for NxTvuAt
N V M B E R s from i to loooo.
II. A Canon of LOGARITHMS of SiMssand
Tangents to every Degree and Minute of the ^adrtmt;
III. A Canon or Table oi Logtjiicd LOGARITHMS.
All which are digefted in a New and moji Compendious
Form, and very corred.
LONDON:
Printed in the Year MDCCXXXIX.
fa I
A TABLE of LOGARITHMS of
NATURAL NUMBERS from i to loooo,
w
I
oooooo
041393
322219
491362
612784
707570
785330
851258
908485
959041
2
301036
079181
342423
505150
623249
716003
792392
857332
913814
963788
i
4
602060
146128
380211
53 '479
643453
73*394
806180
869232
924279
973128
017033
©56905
093422
127105
158362
187521
214844
240549
2648 I 8
2878O2
o
I
2
3
4
1
7
8
_9
lO
II
12
'3
'4
\l
«7
«8
12
zo
ZI
22
23
24
2?
26
«7
28
!2
30
3«
32
33
H
11
37
38
39
oooooo
301030
602060
698970
7781st
845098
903090
954242
477121
M3943
361728
&
724276
799340
863323
919078
968483
oooooo
041393
0791 81
"3943
1461 z8
176091
204120
230449
255272
!Z?754
301030
322219
342427
361728
38021 1
397940
4«4973
43 "364
447»58
462J98
477I2'
491362
505^50
518514
$3 '479
544068
556302
568202
579784
591065
004321
045323
082785
117271
I492I9
178977
206826
232996
257679
281033
303196
324282
344392
363612
382017
399674
416640
432969
448706
463893
478566
492761
506505
519828
532754
545307
557507
569374
580925
51; 21 77
008600
049218
086360
120574
152288
181844
209515
235528
260071
Z83S01
012837
053078
089905
123852
155336
1 8469 1
212188
238046
262451
285557
30^351
326336
346353
365488
383815
401400
418301
434569
450449
465383
480007
494155
507856
521138
534026
546543
558709
570543
582063
593286
307496
328380
348305
403120
419956
436163
45 1786
4^6868
481443
495544
509*202
522444
535294
547775
559907
S717-J9
583 •99
5943:^3
309630
330414
350248
369216
387390
404834
421204
43775'
453318
468347
482874
496930
s 10545
523746
S36ss«i
549003
561101
572872
58433'
595496
A a
♦4 Thi Logarithms rfNaU Numbers to 400.
o
1
2
J
4
I
7
8
9
10
II
12
13
«4
«5
16
«7
18
12
20
tl
Z2
23
^i
26
*7
28
!9
30
J«
3*
33
34
35
h6
37
38
39
69^70!
176091
397940
544068,
740363
,et»9i3
875061
9*94« 9. 934498
97773^4
021189
060698
096910
130534
161368
190332
217484
243038
267172
290035
31I7S4
33243^
352182
37«o68
389166
406540
423246
4393S3
454845
469822
484300
4983 n
5U883
5*5045
537819
550228
562293
574031
585461
1 596597
778iS»
2041 so
4»4973
748188
8 « 9544
880814
982271
025306
064458
100370
'3^539
*643S3
193125
220108
2455^3
269513
292256
313867
334454
354108
372912
390935
408240
424882
440909
456366
47"9»
485721
499687
513218
526339
j3907b
55H50
563481
575188
586587
59769s
J_l _!__
845098 ^5090
230449 -
431364
56S202
67^098
755875
826
029384
068186
103834
136721
167317
195900
222716
247973
271842
294466
315970
336460
356026
374748
392697
409933
4265 1 1
442480
457882
472756
487138
501059
5H548
527630
540329
552668
56^666
576341
5877U
598790
•55272
447158
579784
681241
763428
832509
892095
944485
99 1 226
033424
071882
107210
139879
170262
198657
225309
250420
296665
318063
338456
357935
394452
41 1620
428135
444045
459392
474216
488551
502427
515874
528917
541579
553885
565848
577492
588832
599883
954242
278754
462398
591065
770852
690196
838849
897627
949390
995635
037426
075547
11059a
143015
173186
201397
227887
252853
276462
298853
320146
340444
359835
378398
396199
413300
429752
445604
460898
475671
489958
503791
517196
530200
542825
567026 :
578639
589950
600973,
Tie Logarithms of Natural Numbers t& 1400.
90
fi
92
n
94
95
9^
97
too
lot
102
10^
104
105
107
108
109
no
HI
llSk
119
114
116
"7
118
1x9
120
121
\zz
129
124
1*5
126
127
|28
129
130
131
133
134
135
13^
«57
f3S
I
954»42
472s
9041
9518
96.378«
4260
84«J
«9S0
97.3121
3590
7724
8183
982271
2723
6772
7219
99.1226
■6^9
$«3^
6074
oaoooo
0434
4321
4751
8600
9026
Ol.»«37
$259
703?
7411
02.1189
1603
5306
S7I5
93»4
97R9
03.3424
3826
7Ai6
7825
04.1393
1787
5323
S7I4
9218
9606
o<.3078
34^3
6905
7286
06.0698
1075
4458
4832
8iS$
8$57
07.188*
2250
<U7
5912
91S1
9543
08.2785
3144
61615)
6716
990$
0158
09.3422
3772
6910
7257
10.037 1
0715
3804
4146
7210
7549
11.0590
0926
3943
4277
7271
7603
I2 0S74
0903
3852
4I7»
7105
74»9
X3.0334
0655
3539
3858
6721
7037
9879
0194
14.3015
3327
2
5206
^995
4.731
9416
4051
8^37
3«7^
7666
2111
6512
086S
■5180
94s J
3680
7868
9016
6124
0195
4127
8223
2l8l
6105
9993
3846
7666
1452
sao6
892S
2617
6276
9904
3503
7071
061 1
4192
7(S04
io'>9
4487
7888
1262
4611
7934
1231
4504
7752
0977
4x77
7554
050^
3<^39
5^88
0471
$202
9882
4$I2
9093
1616
8113
2553
1301
5609
987^
4100
8184
2428
<^S33
0600
4628
Uzo
2575
6495
0380
4230
8046
r829
9298
*985
6640
0266
3861
7426
0963
4471
7951
1403
48^8
8227
il?8
4944
8265
1560
4830
8075
1298
4496
767c
0821
395 »
_4
6168
0946
5672
»347
497^
9548
4077
8559
2995
73«£
1734
503S
0300
4520
8700
2841
6942
lOOl
5029
9017
2969
6885
0766
4613
8426
2206
5953
95^8
3352
7004
o526
42.9
7781
1315
4820
8297
»747
5169
«565
1934
S178
859^
t888
515^
8399
1619
4814
7987
1136
4263
__5
6649
I421
6142
0812
5432
000 3
45^7
900 s
345^
7823
116
6^66
0724
4940
91 16
3251
73')0
140S
54iO
94H
3362
7275
1152
499^
88u5
2582
6326
0038
37'8
7368
0987
4575
8136
1667
5109
2090
5510
8903
t270
5610
8926
2216
5481
8722
1939
5133
8503
1450
4574
7118
1895
6611
!27^
589«
0457
4?'?7
9450
3«77
8i59
259^
<5894
1147
53^0
9552
l66^
7757
1812
5830
9811
7^07
2369
7080
1740
63S0
09)2
54'-^^
9895
4317
8^95
3029
7321
1570
>779
9947
40/5
8164
2216
5229
0107
3755
76^4
1538
5378
9185
2958
6699
0407
4085
ZZil
1347
4934
8490
2018
1^518
8990
2434
585 J
9241
2605
5943
9Z5<5
2543
5806
9045
2260
545 J
8618
1763
4885
J^\.^
4148
8053
1924
57^0
95^3
3333
7071
077^
4451
8094
1707
5191
8845
2^70
66
9335
2777
^191
9578
2940
6176
9586
2871
5131
9357
2580
5768
5934
2076
5196
8086
2843
7548
22<y
6808
1365
5875
0339
4757
9130
3460
7748
1993
6197
0351
448<^
8571
£619
66^0
0602
4540
8442
4309
6142
994*
3709
7443
1145
4816
8457
2067
5<547
9198
2721
6215
9681
3119
<553i
9916
3275
6608
9915
5.198
6456
9^90
2900
6o%6
9249
2384
5507
85M
8or6
2fttftfj
72^6
18x9
07S3
$19^
95 £5
3891
8174
i4i5
661^
0775
4896
8978
30&1
7028
0998
493*
8830
26^4
6524
C320
4083
7814
i5»4
5181
8819
2426
6004
9552
3071
5562
0016
34^2
6870
0253
3609
6940
0245
3525
6781
GO 12
3219
6403
9564
2702
5818
lie Logarithms of Natural Numbers to 1900.
*7
9
N».'
I
6438
2
6748
3
7058
4
7J«7
5 1
7576
6
7985
7
8^94
8
140
14.^128
8603 ^99^
141
9219
9527
983 s
01 42
0449
0756
1063
1370
1676
i98£
143
i^.i288
2594
«900
3205
3 J 10
3815
♦ "9
4424
47^8
5032
143
OS*
5640
<943
<S24<S
^549
6852
7154
7457
7759
8061
144
8362
S664
896 5
926^
9567
9%8
0168
0468
0769
^068
'4")
1^.1368
1667
19*7
2266
2564
2^53
3.6.
3460
5757
4055
146,
4353
4550
4947
524*
554«
5838
61 34
6430
6714
7022
»47
7J17
7613
7908
8203
8497
8792
9c85
9380
9674
99«
148
X 7.026 1
^555
0848
1141
»434
172^
2019
2311
2603
2«95
149
15©
ix%6
347«
5381
37<9
6670
4060
6959
4351
7248
4641
7536
4931
7825
52X&
8113
55U
8401
$802
6091
86^9
«5«
8977
9264
9552
9839
0125
0413
C699
0985
1272
»55«
152
18.1844
2129
2415
2700
2985
}270
3554
3839
4123
4407
»53
4^91
4975
i^l^
5542
5825
6io8
6391
6674
^956
7239
»U
7521
7803
8084
%166
8«f47
8928
9209
9490
9771
0051
J5$
19.0331
0611
0892
1171
145 1
1730
2010
2289
25<2»7
2846
»5tf
3125
3403
3681
3959
4*37
4514
479a
5069
5346
5525
>57
5909
5176
6452
6729
7005
7281
755<5
7832
8107
8382
X58
8657
893*
9206
9481
9755
0029
0303
0577
08 $0
IJ24
IJ9
i6o
ao.1397
1670
4391
1943
46^2
2216
4933
2488
5204
275 J
5475
3033
5745
3305
6016
3577
62B6
384^
4110
655^
idi
6%p6
7095
7365
7634
7903
8172
8441
8710
8978
9247
i6»
9$»5
9783
0051
0318
0586
0853
1120
1388
16S4
1921
163
2I.ZI88
2454
4720 2986
3252
35'8
3783
4048
4314
4579
1^4
'4844
5109
5373 5'i38
590a
6166
<S42o
^594
6957
7221
i<55
7484
7747
8010 8273
8535
879?
9060
9322
9584
984^
I6tf
22.0108
0370
0631
0892
1153
'4'4
1675
J936
2lp6
2456
i«7
2716
2976
323<5
3494
3755
4015
4274
4533
4792
50$i
U8
$309
5568
5826
6084
6342
6600
6858
7II5
7372
7630
169
7881
8144
8400
8557
8913
9170
9426
9681
9938
0193
170
2J.0449
07Q4
0960
1215
1470
1724
1979
2233
248 a
2742
171
899(5
3250
3504
3757
4011
4264
4517
4770
5023
5276
17a
15528
5781
6033
6285
5537
678?
7^41
7292
7544
7795
t73
8045
8297
8548
8799
9049
9299
9559
9799
0050
0300
174
24-0949
0799
1048
"97
1546
»795
2044
2295
2541
2790
»75
J038
3286
3534
3782
4030
4*77
4524
4772
5019
^266
I7tf
5513
5759
6006
6252
<5499
<745
6991
7235
7482
7728
"Z
7973
8219
8464
8709
8954
9198
9443
9687
9932
0176
178
25.0420
0664
0908
"J*
»395
KS38
1881
2125
2367
2610
179
2853
309tf
3338
3580
J822
4064
4}0tf
4548
4790
5031
180
5272
55 '4
V'^''
599<J
62zi
<J477
<57i8
6958
7198
7439
181
7679
7918
8158
^398
8637
8877
9116
9355
9594
9833
181
itf.oo7i
0310
0548
0787
1025
12^3
1501
1738
197^
2214
183
M5I
2688
2592
316a
li99
3<J3<5
3873
410P
43^5
4582
184
4818
5054
5290
5525
S7<5i
59»«
6tzi-
6467
^702
(5937
185
7172
7406
7641
7875
8110
nu
8578
8812
904^
9279
186
9^n
974<5
99?o
0213,
044<5
0679
091a
XI44
1377
1609
187
17. 184a
2074
2306
2538
2770
3001
3233
34^4
096
3927
188
4158
438914620
485c
1081
53"
5542
5772
6002
6232
189' ^4^2 ^6691 1^921 ' 71 SI ' 7?8o
-2^
7«?,8
^0^7
8296
8525
i^^riir^^#
\^
The Logarithm of Natural Numbers to 3400. "]
/No.
I
2
3
4
-1^
6
7
-.8 1 9 1
290
46.2398
2548
2697
2847
2997
3146
3296
3445
3594
3744
291
3893
4042
4191
4340
4489
4639
4787
4956
5o»5
5234
29a
5383
5532
5680
5829
5977
6126
6274.
6^23
6571
6719
293
6868
7016
7164
7312
7460
7608
7756
7904
8052
8200
294
8347
8495
8643
8790
8938
9085
9233
9831
9527
9675
295
9822
9969
0116
0263
0410
0557
0704
08s I
0998
1145
296
47.1292
1438
1585
1732
1878
2025
2171
2317
2464
2610
297
2756
2903
3049
3<95
334>
3487
3633
3779
3925
4070
298
4216
4362
4508
4653
4799
4944
J090
iV'J'
5381
5526
299
5671
5816
5962
6^07
62^2
6397
6j42
6687
6832
6976
300
7121
7266
^74<"
7555
7700
7844
7989
8133
8278
8422
301
8566
87 «i.
8855
8999
9«43
9287
943'
9575
97 '9
9863
302
4^.0007
0151
0294
0438
0582
0725
0869
1012
1156
1299
303
H43
1586
1729
1872
2016
2.59
2302
2445
2588
2731
304
2874
3016
3159
3302
3445
3587
3730
3872
4015
4'57
30J
4300
4442
4584
4727
4868
5011
>'53
$295
5437
5W9
306
5721
5863
6005
6147
6289
6430
6572
6714
6855
6997
307
7138
7280
7421
7563
7704
7845
7986
8127
8269
8410
308
8551
8692
8833
8973
9114
9fi5
9396
9537
9677
9818 .
309
9958
0099
0239
0380
0520
0661
0801
09ii
108 1
1222
310
49.1362
1502
1642
1782
1922
2062
2201
2341
2481
2621
3««
2760
29<5o
3040
3'79
3319
3458
3597
3737
3876,
4CI5
312
4»55
4294
4433
4S7*
4711
4^:50
4989
5128
5267
5406
3'3
5544
5^83
5822
5960
6099
6237
6376
6514
6653
6791
314.
6930
7068
7206
7344
7482
7621
7759
7897
8035
8173
3«|
8311
8448
85^6
8724
8862
8999
«"37
9275
94' 2
9550
3^6
9687
9824
9962
0099
0236
0374
0511
0648
0785
0922
i^7
50.1059
1 196
1333
1470
1607
'744
i88o
2017
2154
2290
318
2427
2564
2700
2837
2973
3109
3246
3382
3518
3654
322
379"
3927
4063
4' 99
4335
4471
4607
4743
4878
5014 .
320
, S^S^
5286
5421
JSS7
5692
5828
5963
6099
6234
6376
321
6505
6640
6775
6911
7046
7181
7316
745'
7586
7721 .
322
7856
7991
8125
§260
839s
8S3C
8664
8799
8933
9068 •
323
9202
9337
947"
9606
9740
9874
0008
0143
«277
041 1
324
51.0545
0679
0813
0947
1081
1215
IJ48
1482
1616
175©
32s
1883
2017
2150
2284
2^17
2SS'
2684
2818
2951
3084
326
3218
335"
3484
3617
375°
3883
4016
4149
4282
44' 5.
3*7
• '4548
4680
4813
4946
5079
52W
5343
5476
6800
C609
5741
328
5874
6006
6139
6271
6403
6535
6668
6932
7064
329
330
7196
7328
8645
7460
8777
7592
8909
77*4
9040
7855
9171
7987
9303
9434
8251
9565
828. •
9697
8514
331
9828
9959
0090
0221
0352
0483
0614
0745
0876
1007
332
52.1138
1269
1400
'530
1661
1792
1922
2053
2i!>3
2314 ■
333
2444
2575
2705
2835
2966
3096
3226
3356
3486
$616
334
3746
3876
4006
4136
4266
4396
452-6
4656
478^
4>i5
33S
5045
5'74
5304
5434
5563
5692
5822
595'
608 1
6210
336
6339
6468
6598
6727
6856
6985
7114
7243-
73,2
7501
*^l
7630
7759
7888
8016
8145
8274
8402
8,- J I
8660
87s
338
8917
9045
9*74
9302
9434
95S9
9687
9815
9943
00 ; 2
^ii.
53.0200
0328
04.56
0584 o7t2lo84o'o96.8Uo95l 1223]
liSJ
340
34>
34a
?43
344
346
3.47
348
349
350
3$i
352
353
354
356
357
358
359
360
36.
362
363
364
365
366
,3^7
368
369
370
371
372
•373
'374
'37S
376
377
378
37?
380
381
382
383
384
385
386
387
388
389
fhe Logarithms of Natural Numbers to 2900.
,..-r, , 1607 1734 1862
2754 2882 3009 ^136
4026 4f53 4280 4407
5294 5421 5547 5674
6558 6685 681 I 6937
7819 7945
9076 9202
54.0329 0455
1579
2825
4068
5307
6543
7775
9003
J5.0228
1450
2668
3883
5094
6302
7507
8709
^9907
50,iroi
2293
3481
4666
5848
7026
1704
2950
4192
U31
6666
7898
9126
0351
1572
8202 8319 ^436 8554
9374 949 « 9608 9725
57.0543 0660 0776 0893
1709 1825 1942 2058
2872 2988 3104 3220
4<53i 4147 4263 4378
i\ i^^l 54^9 5534
^34< 64.56 6572 6687
74^92 7607 7721 7836
8039 8754 8868 8983
9784 9898 0012 0126
58.0925 i©39 1153 1267
2063 2177 2291 2404
.3199 3312 3425 3539
433» 4444 4557 4670
5461 7573 5686 5799
0587 6700 6812 6925
^n
8071
9327 9452 9578
0580 ' "
1829
3074
4316
5554
6789
8021
9249
0473
1694
2790 1 291
4004 4r26
52M 5336
6423 6544
7627 7748
882S 8948
0026 0146
1221 1340
2412 2531
3600 3718
4784 4903
5966 6084
7144 7262
8197
1990
3263
4534
5800
7063
8322
21 17 2245
Jill
0705
1953
3*99
4440
083010955
207812203
3323; 3447
,,^- 4564:4688
5678 I5802 15925
6913 703617159
8144 8266:8389
9371 9494 1 96 I 6
'^'■'^'" 0717 '0840
5927 6053
V^2 73*5
8448 8574
9703 9829
1080
2327
357«
0595
1816
3033
4247
5457
6664
7868
9068
0265
H59
2650
3837
5021
6202
7379
88^! 7823 7925] 8047 8160 8272
9279 9391
8944 9055 91& , ,^ ^^,.
9950 00 61 0173 0284 059 6 0507 0619
1938
3»54
4368 4489
5578
6785
7988
9188
0385
1578
2768
3955
5139
6320
7497
8671
9842
lOlO
2174
3336
4494
5650
6802
7951
9997
0240
1381
2518
3652
4783
5912
7C'37
2060
3276
5699
6905
8108
9308
0504
1697
2887
4074
5257
6437
7614
8778
9959
126
229,1
3452
4610
5765
,6917
8066
9212
0355
3518
4787
4812
6049
7282
8512
9739
0962
2181
3397
4610
5820
7026
8228
9428
o6i4
1817
3006
4192
5375
6555
7732
8905
0076
^243
2407
3568
5880
7052
8181
9326
6024 6137
7 "49 7262
2372
3645
4914
6179
744«
8699
8
9954 <^079
1205
2452
3696
4936
6172
7405
8635
9861
1084
2303
35^9
4731
5940
7146
8348
^9h8
0743
1936
3125
43",
6673
7849
9023
0193
1359
2523
^. , 3684
4726* 4841
5996
8295
9441
-_ 0469 0583
1494 1608 1722
263! -^
aIJ^ •'~'^' •'^^'' '**''^
4890 5009 $122 5235
838^
9503
8496
9614
0730
2500
3772
5041
6306
7567
8825
1330
2576
3820
5060
6296
7529
8758
9984
1206
2425
3640
4852
6061
7266
.8469
9667
0863
2055
3244
4429
5612
6791
7967
9140
0309
1476
2639
3800
4957
6ni
7262
8410
9515
0697
1836
2972
4105
6250 6362
7374 7486
8608
9726
084
2627
3899
^167
6432
7693
8951
0204
H54
2701
?944
5183,
6419
7652
8881
0106
1328
2546
3762
4973
61^82
7387
8589
9787
0982
2174
3362^
4548
5730
6909
8084
9*257
0426
1592
2755
3915
5072
6226
7377
9669
osTI
1950
3085
4218
5348
6475
7599
8720
9838
0953
►la
The Logarithms of Natural Numbers to 4400. |
Nf^
1
I 2
1176 1287
3
4
5 <5
7
8'
9
590 S9-»o<55l
'397
1510
1621 1732
1843
i9S$
2o56
391
2177
22S«
2399
t^io
2621
27 }2
2843
^.954
3064
3175
392
3286
3397
3SO8
3618
3729
^840
395040^11
4171
4282
393
4393
4503
4613
4724
4834
4945
5055
$165
5276
5386
394
549^
"5606
S717
5827
5937
6047
6157
6267
<5377
6487
^^'y
6S97
6707
6817
6927
7037
7146
7256
7366
7476
7585
396
7^9)
7805
79»4
802^
8iJ4
8*43
8353
8461
8572
8681
397
8790
S900
9009
9119
9128
9337
9446
95 $6
9665
9774
39^
9^H3
9992
Old
0210
0319
0428
0537
0646
0755
0864
399
400
600973
1082
2169
1 190
a277
1299
2386
1408
2494
2602
162$
2711
1734
2^19
1843
1928
195 1
3036
iO60
*
401
3»44
3152
3361
H69
3577
3685
3794
3902
401©
4m8
4'^2
4226
4334
4442
4550
4658
4766
4874
4982
5089
5*97
403
5305
5413
5521
5618
573^
5843
5951
6059
6166
6274
404
6381
6489
6586
6704
68 1 1
6918
7026
7133
7240
7348
405
745 s
7562
7677
7777
7884
7991
8098
8io$
8312
Hi 9.
406
«526
863}
8740
S847
S9S4
9060
9167
9*74
9581
9488
407
9S94
9701
9808
9914
0021
0128
0254
0341
0447
0554
408
61.06 .0
0767
0873
0979
1086
1 192
1298
1405
1511
16117
4C9
1723
.8z9
190
204.2
2148
2254
2360
2466
2572
2678
410
2784
2890
2996
3101
3207
3313
3419
3525
<;63o
373^
411
3842
3947
40<3
4159
4264
4370
4475
4581
4686
479^
412
4S97
')Oo3
5108
5213
5319
5424
5529
5634
5740
584$
413
5950
60s 5
6160
6265
6370
6475
6580
6685
6790
689$
414
7000
7105
7210
73^5
7420
7524
7629
867$
7734
7839
7943
41 s
' 8048
8153
8217
8364
8466
8571
8780
8884
89891
416
9093
919^
9302
9406
9514
9^15
9719
^823
9928
0032
417
62.0136
0240
0544
0448
o$5z
o6$6
0760
0864
0968
1 072
418
1176
1280
1384
1488
1592
169$
1799
1903
2007
2^1 IQ
4'9
420
2214
2.318
33')3
242;
3456
2515
3559
2618
3063
2732
3766
2i?3S
3869
2939
3972
3041
4076
3 If 6
4179
3249
421
4z8i
4385
4488
4591
4694
4798
4901
$004
5107
5209
422
.. ^312
5415
«;5i8
5621
5724
5827
5929
6034
6135
6238
423
6540
6443
6546
6648
6751
6^U
695^
70s 8
7161
7263
4'4
7166
7468
757'
7^73
7775
7878
7980
8082
8184
8287
425
8389
8491
8593
8695
8797
8900
9002
9104
9206
9308
426
9410
95U
9613
97^5
9817
9919
0021
0123
0224
0326
427
63.042 >s
0530
0631
0733
0834
0936
1038
1139
1241
1342
428
?444
i54'>
1647
1748
1849
1951
2052
2153
2255
2356
429
430
5427
2558
3569
z66o
367Q
2761
3771
2862
3872
2963
3973
3064
4074
3165
4175
3266
4276
3367
4376
3468
431
4477
457B
4679
4779
4881
4981
5681
$182
5*83
5383
432
^484
55^4
6«;88
5^^$
57^5
$886
5986
6086
6187
6287
6388
433
6(588
6789
6889
6989
7089
7189
7189
73 po
434
7490
7590
7690
7790
88l8
7990
8090
•8190
8i89
8389
435
8489
85«9
8689
^789
8988
9088
9188
9287
.9387
436
94?6
958^
968$
9785
9885
9984
0084
0183
0283
0382
437
<J4.o84i
0581
0(J8o
Q779
0879
0978
1077
H76
1276
137$
43B
1474
1573
1572
1771
1871
1970
2069
2168
4267
2366
.
'4^9
2464 12563]
26i5?
2761 h86Q
2959
3058I3I56
3255135541
•^'^' •^fc •^•^•^» •^^ "^^ •^^ "^^ •^^ •^^ •^^
•^» m"^* - "^^ •^^ •^^ "^^ •tf »
The Logarithms of Natuml Numbers ta 5900. » 1 5
140
HI
542
545
544
545
54^
547
54S
549
5^0
551
55.2
553
554
555
556
557
558
559
$60
5^1
562
563
5(^4
5<J5
5<56
5^7
5(^8
5.70
-571
57i
573
5,74
575
576
577
578
579
5«
581
582
58:?
58<
586
589
731-394
733-197
999
7?4.8op
735-599
73<5,99<^
7^7.192
987
738.781
739.572
740:363
741.152
939
742.7*5
743- 5 lo
744.293
745-075
855
74^.634
747-412
748.188
74f.73^
750.508
751.279
752.048
753.583
754.348
755.112
875
75<5.<53^
757.39<^
758/155
911
759.^68
760.422
761. 17i
918
762.679
763.428
764. •7<f
7^5 669
766. 4n
767.1 %6
7<<8.6:?8
769377
770.115
474
277
079
880
679
47<5
272
067
860
651
442
230
018
804
588
371
153
933J
712
489
a6^
040
814
585
356
893
660
425
189
951
712
472
230
987
743
498
251
003
754
5c»3
251
998
743
487
230
972
712
451
189
555
358
159
960
758
556
352
146
939
730
521
309
096
882
666
449
231
on
790
567
343
118
891
66:^
433
202
970
736
501
2^
027
7S8
548
306
063
819
573
326
078
829
•568
326
072
8f7
S62
304
046
786
5*5
263
438
240
040
838
^35
431
225
018
810
599
388
»75
961
745
528
309
089
868
645
421
195
968
740
510
279
047
813
578
341
103
8^4
624
382
'39
894
649
402
,^•53
903
^53.
400
U7
892
6^6
378
120
860
599
336
7'|
518
320
120
918
715
5n
305
097
1^89
678
467
254
039
823
606
387
167
945
722
498
272
045
817
587
356
123
889
6$4.
417
179
940
700
458
214
970
724
477
228
978
727
475
221
966
710
•;53
194
9U
673
410
1^
79^
598
400
200
998
795
590
384
177
96S
71^7
545
332
118
902
684
465
345
023
800
57^
350
122
894
664
433
200
966
730
494
256
016
775
533
290
045
799
552
303
053
802
550
29-5
041
784
527
268
008
746
484
876j9$6
679 759
480
279
078
874
670
463
2S6
047
836
624
411
196
980
762
543
323
loi
878
^53
427
200
971
741
509
277
042
807
570
332
092
851
6p9
366
121
875
627
378
128
624
370
115
859
601
342
0S2
820
557
560
359
157
954
749
542
355
126
915
703
489
274
058
840
621
401
179
955
731
504
277
048
818
586
353
119
883
646
408
168
927
685
441
196
950
702
453
203
.952
699
445
190
933
675
416
156
894
631
8
037
839
640
439.
•237
033
828
622
414
a05
994
7«2
568
355
1,6
918
699
478
256
0J3
808
582"
^>54
125
895
'66'^
430
195
960
722
484
244
003
760
517
272
025
777
528
278
027
774
5^9
164
007
749
490
968
705
£17
9«9j;
720
519
317
^3.
908
701
493
!!4
8j5o
647
431
215
997
777
556
334
iio
885
6>9
431
202
972
740
506
^72
036
799
566
320
079
836
592
347
100
853
603
35*.^
loi
848
594
3^3
082
823
S64
303
041
77«i
f The Logarithms of Natural Numbers to 6900. * 17
NO.
64.0
1 2 1
383
45 «
•i!9
S»7i655
_8^
722
.9
790
806.180
248
3,6
6ji
8$ 8
926
993
06,
129
'97
264 332
400 1 467
ejz
807.535
603
670
738
805
873
941 C08
076 143
6I3
808.211
278
346
+ii
481
548
616 633
75»
818
6^4
886
953
02,
088
•55
223
290! 358
425
49»
5:i
.809.560
627
694
762
829
896
9631031
09S
,65
810.232
300
367
434
501
568
636 ; 703
770
•837
^:?
904
971
642
038
106
"73
240
307
374
440
508
811.575
709
776
843'
910
977
0^4
III
178
6co
812.245
Hi
980
22i
047
i4S
114
ill
,80
■HI
646
3'4
7' 3
38,
780
447
846
S'4
913
6s,
813.581
648
7H
7.8'
848
914
981
048
'i^
187
652
814.248
3«4
381
447
514
580
647
7H
780
847
653
9'3
98©
046
113
'79
246
3,2
378
445
'75
654
815.578
644
7,0
777
&43
9,0
976
042
109
655
816.241
308
374
440
506
Sll
639
70s
771
838
656
904
970
036
102
169
235
30,
367
433
499
657
817.565
632
697
764
830
89b
962
028
C94
160
658
818.226
292
358
424
490
556
622
688
754
8,9
^M.
8«S
610
017
675
083
74>
149
807
i'5
'^73
281
939
346
CO4
412
070
%
gg?
8'9.S44
661
820.201
267
333
398
464
S30
595
661
727
792
662
858
924
989
644
©S5
120
,86
«Si
317
382
448
663
821.5,3
579
710
775
84,
906
972
037
,03
664
8Z2.168
233
299
364
430
495
560
626
C.91
756
665
82^
887
952
0,7
083
,48
2,3
279
344
409
665
823.474
539
605
670
735
BOO
865
930
996
o6i
667
824.126
191
256
32,
386
4J>
516
581
O46
711
668
„ 776
84,
906
97,
036
lOI
,66
231
296
361
669
8*5.426
491
^
621
6S6
Z5i
8,5
W6O
945
010
670
826.075
,40
204
269
334
399
463
778
593
658
671
„ 72«
787
852
9,7
98,
046
III
'75
240
305
672
8*7.369
434^
498
563
628
692
757
821
.86
950
^73
828.0,5
080
,44
209
273
338
402
466
531
59$
674
660
724
789
853
918
982
046
III
«75
2^0
675
829.304
368
432
497
56,
625
690
7S4
8i8j88i
676
947
oil
075
•39
204
26»
332
396
460 .;24.
677
830.^589
653
717
>8.
^ii
909
973
037
102
166
678
831.230
294
358
422
486
55°
614
678
742
'806
67?
680
870
934
573
998
637
c62
7C0
,25
7^4
ni9
828
253
092
317
956
381
019
145
083
832.509
68t
833.,47
2,1
275
338
402
466
5iO
593
657
721
682
784
848
912
975
039
,03
166
230 293
357
993
627
683
834.42,
484
548} 61 I
675
738
802
S66
929
^b
835.056
,20
ii!3iZ46
3,0
373
437
5C0
564
^}i
o^^9i
7H
8i7i88i
944
C07
07,
134
'97
261
68fr
836.324
387
45 J
5H
577
640
704
767
830
8Qi
687
« 957
020
0X3
M6
209
273
336
399
462
f s6
6B8
837.588
652
715
77^
841 904
967
030
093
689
838.2,9
Z82
ill
40S
47» S34|597
660
723
786
• c
•i8
78* Legaritbms of Natural Numbers to 7400.
^
N°.
I
2
3
_4,
_5_
li-
JL\±
_9_
690
838.849
91a
975
038
lOI
164(227
289
352
415
691
8J9.478
541
5o4
667
7*9
792
H^
918
981
043
692
840.10^
169
232
294
*c^7
420
482
545
5o8
671
693
733
795
850
921
984
046
109
172
234
297
694
841.359
422
485
547
610
672
735
797
860
922
^95
984
047
no
172
235
297
360
422
484
547
696
842.609
672
734
7p6
859
921
983
046
^9
108
170
697
843.23 J
295
3^7
420
48*
544
tfo6
731
793
698
855
918
980
042
104
166
229
291
353
41$
699
700
.844^77
539
160
60 1
222
663
284
34<S
788
408
850
470
912
532
22*
594
036
556
H45.09?*
701
718
780
842
904
956
028
090
151
2IJ
275
70i
846.337
399
4*5 1
5*3
584
645
708
770
832
»95
703
955
017
079
141
202
264
326
388
449
511
704
847-573
534
696
758
819
88i
943
004
067
"7
705
848.189
2U
31*
374
435
497
559
620
682
743
706
805
856
928
989
051
II*
174
256
296
358
707
849419
481
542
604
665
726
788
849
911
97a
708
850.033
095
156
^'7
V9
340
401
462
5*3
585
709
646
707
769
830
891
952
014
0^5
135
197
710
851.258
319
38«
442
503
564
6Tj
686
747
808
7"
870
931
992
05J
114
'75
23«S
297
3^!
419
-
71a
852.480
541
602
66^
7»4
785
846
907
968
029
713
853.089
150
211
272
33 J
394
455
516
576
6J7
714
. ^98
759
820
881
94*
002
063
124
ia4
245
7'S
8543o«5
3«7
427
4S8
549.
5io
670
731
792
852
716
. 913
974
0J4
095
156
216
277
337
398
459
7»7
855.519
580
540
701
761
822
882
943
003
064
71S
855.124
'!^
i*'>
306
356
427
487
548
608
668
720
' 729
789
393
850
453
910
970
574
OJI
634
091
694
£5"
754
XI2
815
*7«
875
857.332
721
935
995
056
116
176
236
296
357
4«7
477
722
858.537
597
657
7l8
77»
838
898
958
otS
079
723
859.138
198
258
318
37«
418
499
559
619
679
724
739
798
858
9«8
978
038
098
»58
218
278
72 s
8^0.338
398
458'
5«8
578
tf37
697
757
817
877
726
937
996
o$6
116
176
a3<5
295
355
4>5
475
727
851.534
594
654
714
773
833
893
952
012
072
7l8
862.131
191
251
310
370
430
489
549
608
668
729
7JO
727
787
?82
846
442
9o5
50T
966
561
025
6io
085
680
144
739
204
798
263
85s
853.323
731 917I
977
oj6
096
155
214
274
333
392
452
732
864.511
570
63«
589
748 808
867
926
985
04s
733
85s. 104
163
222
z82
34» 460
459
S18
578
637
73-1
595
755
814
873
9J3 992
051
no
169
228
735
266.287
346
405
465
524
583
641
701
760
819
73«
878
?37
996
055
114
173
23*
29,
350
409
737 i
?67-4<S7
5s6
585
^44
703
7.52
821
880
939
997
7?8Stf 8.056 1
"5
174
533
29*
J 50
409
468
527 58«J
--0I
«S,.4'-
T^■^,
167
R21
870
528
007
o<6
tia.
.7,
740
741
742
743
The Logarithms of Natural Numbers to 7900. *I9
869,222
818
870.404
989
371-573
1872.156
/T- 739
747 873.321
->iO 902
8 74-482
750 875.061
751 640
752 875.218
7§3 795
754 877.371
755 947
7$6 878.522
757 879096
758 669
744
745'
746
747
748
749
759
880.242
76d 814
761 881.385
762 955
763 882.524
764 883.093
765 661
766 884.129
767 795
768 885.361
769 9^6
886.491
770 ^,-
771 887-054
77* ^,J^7
773 888.179
774 «„ 741
775 889.302
776 ^^'^
777 890.
778 ^
780 892.
781
, 782 893
783
784 894
785
786 8f5
787 -
788 896.
789 897
98c
651
.207
762
316
870
'422
.975
.526
077
2;;0
877
46Z
047
631
215
797
378
960
540
119
698
276
853
429
004
579
153
716
199
871
442
012
581
150
718
285
852
4i8
983
547
III
674
236
797
358
918
477
035
593
MO
707
262
817
371
9*5
478
030
581
131
5^4
170
755
339
923
506
088
669
250
830
409
987
564
141
717
292
866
440
013
^
156
727
297
866
434
002
569
135
700
20'5
829
392
516
077
638
756
3H
872
428
985
540
094
648
201
754
306
857
407
642
228
813
398
981
564
146
727
308
887
466
022
198
774
349
924
497
070
642
213
784
354
923
491
059
625
191
757
321
is]
448
oil
573
694
812
370
927
484
040
595
150
704
2^7
809
361
912
462
8
701
287
872
456
040
622
204
785
366
945
524
102
680
256
832
407
981
555
127
69?
270
841
4U
980
548
»>5
682
248
8,3
3_78
942
505
067
629
190
■750
309
,868/
'426
■983
540
096
;65i
205
!759
312
,864
416
967
517
760
345
930
515
098
6$i
262
843
424
003
58I
160
737
889
4^4
038
612
185
7je
328
898
468
036
605
172
739
305
870
.^4
99§
561
123
685
246
806
365
924
482
039
595
151
706
261
814
367
919
47"
022
S72
•20 Toe Logarithms of Natural Numbers to 8400. j
N".
-Ll-iL
i.
4
5
5
7 ^
J_
7;--'
^97.607
682 737
792
8+7
902
957
012 067
122
791
89^.17^
2311235
341
396
451
505
55i 5i5
670
79*
725
780 1 83 5
890
944
999
054
109
•54
218
793
8^9.27?
328
?83
437
492
547
5o2
555
7"
766
794
820
87')
9>o
985
039
09+
149
203
V^^
312
795
900.357
422
476
551
58^
540
595
749
804
858
796
9l^
968
022
077
131
166
240
295
349
404
797
9M.458
5M
567
622
676
731
785
840
894
948.
79'J|90i.oo3
057
112
166
229
275
329
384
438
492
799
547
601
144
<555
198
7«0
253
7h
307
818
35i
873
415
927
470
9ii
524
036
578
800
,902.090
Soil 632
587
74 «
795
849
90J
958
012
o5tf
120
^21934.174
228
283
1^7
391
445
499
553
5o7
(56 1
805
715
770
824
873
5>32
986
040
094
M8
202
804
905.256
310
j54
4t8
472
S25
580
634
588
742
805
795
8so
904
9$8
012
o(5$
119
173
227
281
3o6
906.33$
359
44?
497
ISO
5o4
558
712
755
820
07
H73
927
9ii
03$
089
142
195
250
304
358
^08
907.41 1
455
5«9
573
626
680
734
7S7
841
895
809
810
94}^
002
539
056
592
109
646
163
69)
217
753
270
.807
324
860
378
914
11}
967
908.485
3ii
909.021
074
128
lU
235
2S8
342
395
449
502
812
555
609
6<$3
7«6
770
823
877
930
984
037
l'^
910.090
144
197
251
304
358
41 1
464
518
571
S14
624
578
73f
784
838
891
944
998
Oil
104
?'^
911. 158
211
2i54
?I7
371
424
477
550
584
637
S16
690
743
797
850
903
955
039
053
Ii5
169
817
912.222
27$
328
38i
425
488
541
594
547
700
S18
7U
8o5
8lP
913
966
019
072
I2S
»78
231
819
820
913.234
127
867
390
920
443
97?
49«S
026
_S_49
079
5o2
131
^55
184
703
*37
761
290
814
821
914-543
872
395
449
501
555
608
55o
713
765
819
822
925
977
030
083
135
189
241
294
547
8.3
91^.400
453
505
$58
611
644
7i5
759
823
874
824
927
980
033
085
138
191
*43
296
349
401
82$
916.454
507
559
612
664
717
770
82 i
875
927
Si6
9^0
033
0^5
IJS
190
243
295
348
400
45 3
827
917-505
SS8
610
(S63
7»S
758
820
873
9^5
978
828
91&.030
0S3
135
i38
240
292
345
397
450
502
8^9
5'?4
607
660
712
764
816
869
92c
973
oz5
S?o
919.Q78
130
183
235
287
340
392
444
496
549
831
601
653
70s
758
810
852
914
957
019
07 »
832
920.123
175
228
2S0
332
384
435
849
541
593
8i?
645
691
749
801
853
906
958
010
062
114
'^'?4
921.165
318
270
322
374
425
478
530
582
534
83<;
686
738
790
342
89+
945
998
o$o
102
154
835
922.206
258
310
362
414
455
518
570
622
574
837 725
777
829
881
933
985
037 088
C40
192
S38 923.244
296
348
399
451
S03
555 <5o7
658
710
S39 162 814I865I
917 969lo2ilo72li24li76"288|
191
m
_ . . 5^5®^^* ;#! tea :5Sf "~ "'
T<33n^ii^KHg^^|^^e?| rSia^ LOfiL tfi^^
•^fc a^fc «^fc •^k "^^ •^^ "^^ •^^
* 22 Hie Logarithm of Natural Numbers to 9400.
890
891
892
893
894
895
896
897
S98
899
900
901
902
903
904
905
906
907
908
y>9
910
911
912
913
914
915
916
917
918
919
920
921
922
925
924
92$
925
927
928
929
930
931
932
953
93+
9^')i
949.390
950.3(^5
851
951.337
28g
9^1.308
792
953275
' 760
954-242
725
955.205
688
956.168
649
957-128
507
958-086
564
483
950
969.416
S82
970.347
812
936.971.276
937! 743
938 1 972.203
939 1 665
I
2
3
4?9
4t8
53«
9i6
975
024
4'?
463
$"
900
949
997
3»6
43 s
483
872
920
9<9
35«
40$
453
841
88p
938
3»5
373
421
80H
9$6
90-;
391
339
387
77J
82 1
869
aS5
303
351
7?«
784
8J2
2I<J
2<4
312
fij>7
744
792
176
224
271
6?5
703
751
»34
181
229
61 a
659
707
084
'37
1S4
$65
6.4
661
042
090
138
518
56S
6'3
994
041
089
469
5.6
563
943
990
033
417
464
5"
^o
937
985
36?
410
457
8J5
8S2
919
307
354
401
778
825
872
249
296
343
719
766
813
189
235
*83
6si
705
7$2
127
173
220
595
642
588
0(J2
109 1^6
^30
576 523
996
043
090
462
509
556
929
975
02 r
39J
443
485
85^
904
951
322
369
415
78^
«3a
879
249
295
342
712
758
804
4
5
6
7
8
58$
634
683
731
780
073
(21
170
219
257
5<5o
608
<557
705
754
045
095
»4?
192
240
532
580
629
tf77
726
017
o56
114
153
211
502
550
599
547
696
985
034
083
131
180
470
518
555
61$
653
953
00 1
049
098
»45
435
484
53*
580
528
918
966
014
o52
110
J^5>
447
495
543
592
880
928
975
024
072
351
409
457
505
553
840
8S8
93<5
984
032
320
358
415
464
51'
799
847
894
942
990
277
32s
373
420
468
755
80J
850
898
946
232
280
328
?75
423
709
757
804
^"^l
900
1*5
233
280
328
376
551
709
7J«
804
851
ijd
184
231
»79
325
5ii
558
706
753
801
08 s
132
180
227
»75
559
606
<S53
701
748
OJ2
079
126
'74
321
504
5S»
599
546
593
977
014
071
ti8
i5<
44s
495
542
590
637
919
965
013
o5o
108
390
457
484
«i
578
85
907
954
OOI
048
329
376
423
470
517
798
845
892
939
986
257
314
351
408
454
735
782
829
875
922
203
249
295
343
,389
570
716
753
810
856
Ij6
183
229
275
323
5o2
549
69$
74»
788
oS\i
114
161
207
254
533
579
625
572
719
997
044
090
137
183
4*1
508
554
5oo
547
9*5
971
oi8
064
Ho
588
434
480
527
573
8<i
«97
943
9«9
1035
_9_
829
315
803
289
774
259
744
228
711
194
«77
158
640
120
5oi
080
559
03»
515
994
47'
947
4*3
899
374
848
322
795
258
741
212
584
155
525
095
554
033
501
9*9
435
903-
359
835
300
7*5
229
693
156
619
082
Tbe Logarithms if Natural Numbers to 9900. •a3|
940
»74
' Z
220
3 4
266 3'3
359
405 451
8
497
-9.
543
973-128
941
590
636
682
728
774
820
866
913
959
ooS
942
974.051
097
•43
189
235
281
327
373
420
466
943
5i»
558
604
650
696
742
788
834
880
926
944
972
018
064
IIO
'A
202
248
294
340
386
945
975'432
478
5*4
570
616
661
707
753
799
845
94«
891
937
983
029
075
121
166
212
258
304
947
976.350
396
442
487
533
579
625
67.
717
762
948
8c8
854
900
946
991
037
083
129
'75
220
949
977-266
3>2
3^8
403
449
495
54«
586
632
678
950
724
769
81S
861
906
952
997
043
089
»35
95'
978-180
226
272
3>7
363
409
454
500
546
S9«
952
<37
683
728
774.
819
865
9"
956
002
047
953
979.093
>38
184
230
275
321
366
412
457
503
954
548
594
639
685
730
776
821
867
912
958
955
980.00}
049
094
140
i*5
23 1
276
322
367
412
956
458
503
549
594
640
685
730
776
821
867
957
912
957
003
048
093
139
184
229
275
320
958
98i.3<J5
411
456
501
547
592
637
683
728
773
959
960
.819
864
316
909
362
9H
407
000
452
04s
497
090
543
'35
588
181
633
226
678
982.271
961
723
769
814
859
904
949
994
040
085
130
962
983-175
220
265
310
356
401
446
49'
536
581
965
626
671
716
762
807
852
897
943
987
032
9<54
984.077
122
.67
212
257
302
347
392
437
482
9«5
527
572
6,7
662
707
752
797
842
887
932
966
977
022
067
112
157
202
247
292
337
3«2
96y
985.42(J
471
516
56.
606
651
696
74'
786
830
96%
875
920
965
010
05s
100
'44
189
234
279
969^
970
98<J.324
369
S16
4>3
861
458
906
503
9S»
548
995
593
040
637
085
682
130
727
1^*
77'
P7I
987.219
264
309
353
398
443
487
532
577
622
972
666 711
756
800
845
890
934
979
024
068
973
988.113 l«S7
202
247
291
336
38.
425
470
^A*
974
559. 603
648
693
737
782
826
871
9»5
960
975
989.COS 049
C94
138
• 83
227
272
3.6
361
fl
916
450 '494
5?9
583
628
672
717
76.
806
850
977
895 939
983
028
072
117
161
2C6
250
294
978
950.339 383
428
472
5.6
56,
605
650
694
738
929
980
783
827
270
87 J
3^5
916
359
960
1°§
004
448
049
492
093
536
L37
580
182
6^S
991.226
981
1669 713
757
802
846
890
934
979
023
067
982
99i.iii 156
200
«44
288
333
377
421
465
509
983
553598
642
686
730
774
818
863
907
951
984
995 039
083
127
172
216
260
304
348
392
985
993.436 480
524
568
612
657
701
H^
789
833
986
877 92«
965
009
053
097
141
'85
229
273
987
994.317 361
405
449
493
537
581
625
669
7»3
988
. 757 801
845
880
933
967
021
064
108
IC2
9^ 99'i-i96 24ol284'328l372l4«6l46o
504
S47>59'|
•24
7 he Logarithms e/I^atural Numbirs to loooa |
N'.
—I— (.» *
3
4
S
6
7
9
9
990
90^ 03J 1679' 723
767
811
854
898'
942
986
030
991
996.0741117
161
205
249
293
336
380
if*
468
992
• S>2 555
599
643
687
730
168
774
818
^62
905
993
949
993
037
oiio
124
212
«55
299
343
994
997.386
430
474
S^7
56.
605
648
692
736
779
995
?23
867
910
9S4
998
041
o8f
128
172
216
996
998.259
303
346
390
434
477
521
564
6o«
652
997
69s
739.
782
826
869
9<3
956 000
043
087
998
999.130
174
2l8
261
305
348
39« 435 478
522
P99 ' <6i; 1
6oQ'6?2'f96|
739
2lL
826 870 913 9571
A
T A B L E
O F
j^tificial or Logmthmjc
SINES and TANGENTS,
To every Degree and Minute of the
Quadrant,
*D
T 'A B L E
1
O P
'Artificial or L^aritbmic
SI N E S,
To every Ibegree and Minute of the
Quadrant.
Min.
o
1
2
i
4
6"
7
8
9
lo
II
12
«3
H
'5
i8
»9
20
oDeg.
6.4637
6.764^56
6.940847
7065786
7*162696
7.241877
7.308824
7.366816
7.417968
7.505^8
542906
1 7.577668
,7.609853
; 7639316
TdbU of Lt ^qjHtbmc Stttei
IS
2 Dcf .
83243
8977s
96207
302546
08794
7.667844
7.694173
I7.718997
'7-742477 1
7764754 1
H954
21027
27016
32924
38753
44S04
50180
|S783
66777 I
8 542819
4999$
S3539
57C'54
60540
65999
67431
70836
66
5^718800
21204
«59?2
8.730688
JflJeg.
f
9715*
8-600932
03489
06623
09734
7667
„ 9969
8^742259
4536
68ot
o 9055
8.751297
3528
5747
8.7601 J 1
«337
45"
8.843584
5J87
7Hi
o»*97l
8>85Q7$i
«24
J*9
7801
9546
8.861 281
3014
4738
8165
9868
8.871565
325
0615
8x85
Min-
II
12
«3
»4
16
>7
18
19
so
t T A B L E
r
Artificial or Logmthmic
TANGENTS,
to tvery t>egree and Wnute of the
Quadrant,
d ^dU ef Logantbtnic frngHas.
L
o
6 Deg.
1 Deg.
»Tkg.
3 r«8.
8.7«939«
4D*
Mill
6
o
8. f 41921
i.543083
8.844^44
t
6.463736
49101
46691
«i8o6
<!4«5
I
a
6.764756
56165
50268
44aQJ
i6588
8a6o
a
i
6.J40»47
6JI15
51817
8.850057
3
4
7.065 7«tf
6995*
^ 573 J<5
«8959
1S46
4
-6
7.^62696
76691
60828
8^31317
3628
5
6
r-24»«78
«33«3
64291
3^6}
5403
I
7;|o882$
89«5«
677*7
599«
7171
1
t
7.366817
96a9«
7UJ7
« *5'7
8932
i
9
7.4»797o
8.302633
745«o
8.74«3*26
8.86C686
s»
(I
I'^iT^l
08884
.15046
77877
Z9t2
2433
Id
II
7.505 wc
' 81208
<ao7
4172
tJ
7.542909
sun
84,14
7479
, 590$
12
:»3
7.577671
27114
87794
9740
7632
13
14
7.6<J9857
33025
91051
8.751989
- « 9'^ '
14
i6
7.639820
38856
94««i
« ,9749*
4227
8.S71064
2773
15
Tift
7Myfi49
44610
<5453
«7
7.694179
50289
8,600677
866ii
4469
]l
i>
7.719003
55895
03839
8.760872-
6162
19
7.742484
61430
06978
3065
7849
19
2o »7.7«47«i
66894
10094
5»45
9529
10
»Pa"^
. ..._...
L,_.
J3
34
36
.39
40
■r —
> 41
-J 4^
«iJ"|9#li'-^^J 44
.«ff .,'##J(5<il 4«
l:|!iif
••«•»••«•» m*^*m V^V VqV «-ff^-9*«-W9
©9 ©Q ^^ ^^ ^^ ©O ©Q
:|:
•,«^»
*";'^,
Mixu
21
22
23
28
29
31
32
BB
94
21
37
38
39
4^
42
43
44
45
46
47
48
49
JO
51
52
55
54
J5
5^
57
5«
59
60
7. 78708 8. 9^5*5 8. .8340
7. 95099 8-403338 8-631308
7.910894 8. 08304 8. 4255
7. 26134 8. 13213 8. 7185
7. 408 <> t 8. 18068 8.64CC93
7. 55100 8. 22869 8l 2982
7. 68889 8. 17618 8, 5853
7- 822 < 3 8. 32315 8. 8704
7. 95219 8. 36962 8.651537
8*0078 09 8. 4156 8^ 4352
J Table rf Logaritbmic Tanggn is.
3l>cg.
_oDeg. f iD^
•25 k,
7.785951 8.372291 8.613189
7.806154 «. 77622 8. 6261
7. 254^ 8- 82889 8. 9313
7* 43944 ^' 88092 8.622343
7> 616 74 8. 93234 8. ' 5352
8. 20044 ^* 46110 8. 7149
8. 31945 8* 50613 8. 9928
8* 43527 8. 55070 8662689
8. 54809 8. 59481 8. , 5433
8. 65806 8.^849 8. 8 160
8^76531 8. 68172 8.670870
8. 8^997 8. 72453 8. 3563
8. 76693 8. 6!239
8,107202 8. 80892 8, 8900
8. 97217
8,107202 w. ^wwy* »/• wyu^j
8. 16963 8. 850 50 8.6815-44
8, 26510 8. 89170 8. 4172
8* 35851 8, 93250 8. 6784
8. 44996 8. 97292 8. 9381
8. 53952 8.501298 8.691963
8. 6*727 8. 05 267 8, 4529
8. 71328 8. 09200 8. 7081
8. 79761 8. 13098 8. 9617
8 88036 8. 16961 8702139
8. 96156 8. 20790 8. 4646
8.204126 8. 24586 8. 7139
8.204126
8. 71953 8. 28349 8' 9618
8. 19641 8. 32080 8. 712083
8. 27195 8. 35779 8. 4534
8. 34621 8. 39447 8. 6972
8. 41921 8. 43084 8. 9396
2Dcg.
8.767417
8. 9578
8.77>^727
8. 3866
8^J995
8. 8114
8,780222
\ 2320
. 4408
. 6486
t. 8554
^.790613
8. 2662
8. 470 1
_67^
8. 8752
8.800763
8. 2765
8- 4758
8.^7^
8. 8717
8. 0683
8. 2641
8- 4589
8. 65 29
8. 846 1
8.820384
8. 229S
8. 4205
8. 610?
8. 7992
8^ 9874
8.831748
8. 3613
8> U7 ^
^ 732J
8. 9163
8.84C99S
8, 2824
I9. 4644
3.881202
8. 2869
8. 4530
8. 61^5
8- 7835
8. 9476
8.891112
8. 2742
8. 4366
^_5984
8. 7596
8, 9203
8.900803
^. 2393
8>_3y87
8. 5570
8. 7147
S. 8719
S.91Q285
8._ 1846
8. 3401
8. 495 »
8. 6495
8. 8034
8. 9567
8,921096
8. 2619
8. 4*36
8. 5649
8. 7[56
8658
93015$
8. 1647
8. 3134
8^_ A6ie
8. 609 i
8. 7565
8. ^ 9031
8. f 40494*
8. 1952
Mia
21
22
23
24
J5
26
27
28
29
JO
31
32
33
34
J5
3^
37
38
39
JO
41
42
43
,44
46
47
48
49
50
5^
52 «
5^
54
J5
5<5
57
58
$9
»yi DggfiBmeTSuSs Dyrnt Sf 6, 7> ^» ^ 'Q» i y> la, Ltgaritbm
5- o
to
io
)o
40
_$£
6. o
to
20
30
40
_!2
7. *o
10
to
Jo
40
_5o
3. o
10
io
30
40
50
h o
lo
10
30
40
50
0. o
lo
JO
40
_ |o
1. o
10
30
40
2. O
10
20
30
40
so'
8.94.019^
95-44W
96.iH9
9».1S?3
W-4497
p.0 070 40
01.9235
C3,io89
04.^^$
05.3&59
06.4^5
07'54gQ
085894
09.6061
10.599*
11.559S
12.5187
'3 44 70
M.35$5
15.24$!
|tf.x664
16.9702
17.807*
t8.628 o
t945}2
202234
9992
21.7609
22.$09£
23.2444
9^70
24.6775
25.37^1
26.0633
7394
^27^094
28.0599
7048
89.3399
9^55
30.5819
Jjji893
7879
32.37^0
9599
33-5337
1148
7«87
4029
0276
6430
24v^
5^473
4366
0176
S906
34.0996 155^
^579j7>34
1738
5894
9600
2883
5768
8*78
0435
:I256
3762
49^
5»85
6^33
69Z2
70165
6973
6656
6I25
5387
4453
3330
2025
0546
8900
7392
5^^
3017
0759
8363
58^3
3<72
0386
7478
4453
1314
8065
4708
3x74
7*84
0947
4189
703^
95JO
1^92
3421
4«95
6071
69^2
7583
7947
8066
7951
76ii
7060
6303
53SO
4^08
2^85
1389
9726
790^
592s
37^7
1526
9ii6
6572
3899
IIOI
8181
5H4
1994
8734
5867
1897
8326
4658
0895
7041
3097
9066
49SO
0753
<^475
2119
7687
JL.
A^5
8670
2289
549*
"99
^1
1825
45Si
6026
7172
8036
!£ll
8970
0065
8921
8$66
7993
7216
^43
3743
22|0
0551
6719
4^77
2291
9«6g.
7^11
1814
888^
5Sh
^73
9402
6024
M44
8964
5286
1514
7650
3698
9658
5534
1328
7043
2679
8240
6034
oo$r
5628
6789
95^0
^26
4016
S741
7M4
Si7l
9*07
9^76
9990
006 1
99<5t
95 «9
892$
8127
71^6
5957
4^00
3070
1374
95»9
7^11
<354
3055
061 g
^448
5849
2516
9589
652^
3351
0069
6681
3190
9600
5913
2132
8259
4*97
0249
6117
1903
7610
3239
8792
5
74^
1429
496*
80^3
o»l6
311^
5205
6895
8»79
93^7
0176
I0d8
1^56
0469
98*4
9037
8026
683d
5414
3908
2196
OJ2J
^04
6131
3818
8784
6o7i
3237
0282
72H
4017
073:5
733_7
3^3^
0235
6539
3748
8867
4896
0840
6700
2478
8176
3797
9343
6
887!
2801
6293
9374
2069
4400
6386
8<^48
9400
0460
124X
1759
90Z4
2047
1842
1417
0781
m4
8915
7700
6307
4744
3G^»6
IK30
6906
4$79
&I15
9518
fm
3947
0980
7M
4703
1399
2991
4480
0870
7164
33^4
9474
5495
1430
7181
5051
8742
43S5
9893
0287
4170
7619
0660
JSt8
5613
7567
9»97
0519
«S5i
2J05
•797
3637
3037
2809
2362
1704
0856
9801
8569
7»59
5578
3834
«933
9879
7679
2861
•252
7519
4^56
8583
5377
2063
8644
5124
1504
7788
3979
0080
6092
2019
7862
3624
9306
491 i
0443
1696
5534
8941
«943
M
8744
fM
4«47
4<WS
J774
Jto6
t6}o
«7S4
o«86
SU
4*51
fZ.34
0666
845*
0984
Sajs
m
60J1
2726
9297
5766
2137
8411
4593
068$
6688
2607
8442
4'95
9871
5469
|<»99
689J
0*59
3222
IS?
99«»
1485
2748
3723
44*4
4864
5056
JO 10
4737.
4248
355 f
1569
0300
8855
7*4*
5466
3$34
»4S«
92XS
6854
4349
1714.
8953
6069
3067
995*
6725
3388
9SM^
6408
2768
9034
J207
1289
J«94
9021
4766
0434
0024
099*1 »S4<*
Logarithmic Tm^0Mts Dggrus^ 5^ 6, 7, 8» 9, ic, :
5. o
10
90
30
40
50
6. o
10
10
30
40
SO
7. o
10
20
30
40
—12
S, o
10
20
30
40
50
P- o
io
20
30
40
_5o
!©• O
10
20
30
40
_?•
II* o
10
ao
30
40
__5o
12. O
10
20
30
40
$0
9^.6267
9^3S77
^.00.9298
OS. 1620
04.5284
€5.6^59
06,7752
07.«57^
o«9«44
09.94^8
ia9559
11.9429
129086
»3-854a
I47^«
I5«77
»<5.5774
'7.4499
1 8.3059
1 9' 1462
9712
20:7816
xi-5779
22.3606
23.1902
8871
24.6319
a5-§<^48
26.0862
7967
974964
2S.|8s8
8652
«9.5349
30.1951,
84^3
3M885
32.1222
7474
9739
34-5755
35.1697
75661
3404
7<573
149^
4893
790y
0546
2834
479«
^434
7781
8846
9^44
0157
04«7
0556
0404
0041
947^
8718
7775
6654
5362
3907
2294
0529
8619
6568
4382
2065
9622
7057
4374
M73
8671
5658
2542
9326
6Q13
2607
9109
55^3
1852
8095
4^59
0344
<^353
2287
8I49
_2
4852
9075
2855
6217
9188
1790
4044
59^9
7582
8900
9938
07 fQ
1228
1504
M5I
1377
0993
0409
9632
8671
7532
6224
475«
3124
»345
9420
7356
5^56
2826
0371
7794
5 ICO
229»
9375
6351
3225
9999
6677
3261
9714
6159
2479
8715
487X
0948
6949
2876
87311
6295
0473
4209
7532
0465
3031
5251
7144
8727
0016
1027
£773
2266
2519
2543
2348
1944
1340
0544
9565
8409
7084
5596
3953
2159
0220
8141
5929
3586
1118
85 JO
5824
300$
0077
7043
3927
0671
7339
39«4
0398
6795
3106
9334
5482
1552
7545
34^5
9313
_4
7734
1866
55^0
884?
'737
4268
6455
8316
9869
113Q
2113
2833
3302
3532
3533
3317
2893
1269
1454
04s 7
9x84
7942
6439
4780
2971
1018
8996
6700-
4345
1865
9164
^547
3717
C779
7734
4588
1342
8001
4567
1042
7430
3733
9953
6093
2155
8141
4053
9893
_5_
9168
3254
6906
0149
3007
5502
7655
9485
1008
2x40
3197
3891
4335
4542
4521
4284
3839
2363
»347
0157
8799
7280
5606
3782
1815
9710
747*
5103
2610
9998
7269
4428
M79
8424
5268
2013
8662
5218
1685
S064
4358
0570
6702
*757
8735
4640
0474
0597
4639
8248
1451
4272
6732
8852
0651
2144
3348
4278
4947
5367
5550
5507
5249
4783
4121
3269
2236
1029
9655
8119
6430
4592
2611
0492
8239
5859
3354
0730
7990
5138
1178
9x13
5947
2682
9322
5869
2327
8697
4983
1187
7311
3358
9329
5227
1053
7_
2021
6019
9586
2750
55^4
7959
0046
1813
3277
4453
5356
60CO
63^95
6556
6491
6211
5726
^044
4171
3133
1899
0508
8957
71^3
54CO
3405
1272
9007
6614
4097
1461
8710
5847
2876
9801
6624
3350
9980
6519
2967
9329
$607
1803
7919
3958
9922
5813
1632
8
3Hi
7394
0920
4045-
^792
9183
I2?7
2973
4407
5556
6432
7052
7422
7559
7472
7172
6666
59S5
5077
4008
2767
1360
9794
f|074
6207
4198
2052
9773
7368
4839
2191
9418
6555
3573
0489
732'
4017
0638
7167
36C8
9961
6230
2418
8527
4558
0514
6398
2210
I
i
*3^ Logarithm Sines^ Peg. 13, 14, 15, 16, 17, 18, 19^ 20. Index 9-
^ t^ar. ^ines^T^. 21, 2X, a 3, 2^ 15^ zbj 27^ a«/ A. 9? '
i~^ ^ 'I'M A|.j_-l| ^ ^ ig^
II. o
10
30
40
22.0
10
30
30
40
_$o
23.0
10
20
30
40
24.0
10
20
30
40
^«o
25,0
10
20
50
40
_5o
26,0
10
20
50
40
27^0'
20
30
40
tS.o
10
20
30
40
4^129
7606
$6.08^4
407 «l
7269
? 7. 043 5
3f;7y
6689
9777
58.2840
5877
^8890
59,1878
'4^42
4658
7932
1178
4396
7587
0752
3888
0084
3144
6179
9190
2175
. ^-,5«37
7783; 8*375
60.0700 [0990
3594|388i
^9313
01.2140
4944
62.0488
3229
~ 4647
63.1326^
; 9^4 2
^^4.1842
6984
^ 95^7
65.2052
^ im
66.1970
9255
67.1609
3977
6328
8663
68.0982
w
6751
9597
2421
5223
8004
0763
3$02
6219
8916
»593
424?
6886
£503
2101
4679
7*39
97«i
2303
4^08
7295
97<^|
2214
4648
7065
§46^
1847
4213
6562
8895
1213
:t
4987
83 $8
I $01
4716
790,
1061
4.200
309
0391
6482
9489
»47S
8368
1280
4170
9880
2702
8z8i
1P38
377f
6490
9184
18^9
4J'4
7148
9764
23S0
49 j6
7494
0034
5057
7541
0009
2459
4891
73PS
gro3
^084
4448
6796
9278
•441
^74*
5J>S
8J83
1^24
?036
Siaz
j38o
45"
7618
t»69P
1753
P7«3
9289
2770
J7*7
8660
1570
4457
L'i'
0163
2983
5781
8558
131J
♦?1Z
6760
9453
2125
: 4778
74>i
0024
2678
S»93
7749
0287
2806
77*0
joijs
2703
mi
7546
994^
2321,
4684
7030
9|6o
W4
1^
597' *99
9«|4|9S58
964*
»909
2146
5356
8|39
l^
4824
7947
1005
4058
7085
0088
3067
6021
8952
i860
4755J
7607
°^
6060
8834
1587
45«9
70J0
S72I
2391
504*
7673
0284.
24(58
5676
8855
2009
8004
0539
3057
5519
8037
050P 07^6
«375
7786
0181
49»9
7*64
959*
96$
4zoi
5136
8236
1312
7386
53«3>
9*44 9SJ6
^03«
789*
0729
6338
9110
1861
4£?^
J7JO0J
A
■ 5506
7935
°544
«87<5j3,35
5450 5706
8258
0793
83Q7
Sfoj
8284
[026
0419
»S5» <7«
7497
.98^
SM35
^9<H
5l»5
9^7*
5447
8|45
4f6j
7^88
•^8$
6626
9883
314 s
4314
94(88
2636
8853
1924
4968
7988
0984
i<l|
HI9
SJ«9
8176
1912
3825
6616
938tf
4861
7570
t«S7
«^23
$70
197
0804
3393
5962
8512
1044
!2f4
8531
0991
8266
:«^S8
$389
f773»
oo(;6
|95S,
6903
9827
27*8
j6o6
8461
"?4
6068
9162
2229
5272
8289
£282
4^S«
7196
0118
30 17'
589*
1745
X57^
4385
7172
9<62 9937
240^
7840
0524
5833
8458
1004
36?o
6218
8766
'297
380^
6302
«778
1236
3677
OlQQ
8506
08^
3268
i6?4
7964
9*«7
4887
^53
0207
9804
2949
637J
94W»,
«S34
SS74
«$9<»
ijSo
4^47
749«
0409
6178
9029
26f2
5406
8109
0792
3i«9^4S4
A097
8720
j3*3
3908
6473
9020
»549
4058
6550
J 9025
*48i
39*o
6341
8746
lit*
S50J
S85S»
! .8197
•i«9
•852
nil
7280,
o$3»
|7>J
6951
012a
326J
1 858
4665
74>'5
0213
2956
1^7,
837«
to$9
16360
8981
1583
4165
*7»9
9*74
i%>o
4308
67P9
9271
»7%5
8986
1123
,F4».
6094
8430
<^So
3o^S
•il*
4» 4» 4» 4» ♦ ♦ ,15^, •ggwe: . *s* *S"
^3o
.^'•-
AMKr,
^'?.
29» 30. 3^
3*' 33' 34, 35. :
j6. /ir
-9. i
29.0
10
So
30
. 40
30.0
10
20
30
40
5^
31.0
10
20
?o
40
50
32.0
10
20
30
40
50
33.0
^0
20
30
40
50
34.0
10
20
?<^
40
50
35»o
10
20
30
40
50
35,0
10
2o
30
40
y
I
$799
»o69
oj«3
2562
4786
•599,
9189
i3''8
S683
7819
9941
2049
414*
6214
8291
034$
tib.
44"
5426
S427
0415
2 J 90
4353
6303
8241
0157
2380
398a
^21
7749
961J
1469
3312
5143
696$
8772
0569
2356
4131
5896
7649
9392
1125
2847
4558
5259
7950
2
6027
829?
0548
278 s
5007
721$
9+07
1585
3749
5897
80}2
0153
a»6o
4352
0432
8497
OJ49
2$8S
4614
66i6
8626
0613
2387
4548.
6498
8434
OJ59
1271
4>7»
6059
7936
98O1
1654
3495
«326
7»44
8952
0748
2534
4^08
6071
7824
9566
1298
3018
4729
6429
8119
»254
8521
0772
3008
5229
743$
9626
I802
3954
6lI2
8*4$
0364
2469
4$6i
6639
870}
0754
2791
4815
6827
8825
0811
2784
4744
6692
8627
0550
2462
4361
6248
8123
9987
1838
3679
5508.
7326
9132
0927
271a
4485
6247
7999
9740
1470
3190
4899
<S598
8287
4 ; 5 1
6
5936
9198
M4*
3*76
5892
^094
ct8o
24$*
4610
6753
888z
0997
3098
5186
7259
93»o
1366
3430
54iO
7428
9422
1404
3373
5329
7274
9205
1125
3032
4928
68 1 1
8683
0543
»392
4229
6054
7869
9671
1464
324$
5015
<5774
8$32
0260
1987
3704
5410
7106
8791 1
_7_
7163
94»3
1668
389*
6119
0498
2669
4825
6967
9094
IZ08
3To"8
5394
7466
9$25
1570
3603
5622
7628
9^ai
1601
3569
74*57
9398
1316
3223
5117
69^
8870
0728
2576
44"
6236
8049
9851
1643
3422
5191
6949
8696
0433
2159
387s
5580
7»75
8959
_«»
7?Sy
9648
189.
itto
^354
8532
0716
2gii5
5040
7180
9306
'4'9
3517
$5oi
7«7t
97JO
»774
3801
$8«3
7828
9^<9
•799
3765
*7«9
76tfl
9590
1507
34«3
5306
7187
905I
0914
»76o
4$95
6418
8230
0031
1821
3599
5367
7124
8^71
0606
4331
4046
5750
7444
9127
9
68.5571
5p.oop8
2339
4$«4
<774
6482
8747
0996
mi
5450
7654
9844
ioip
4175^
6326
8457
0575
2679
4769
6846
8908
09j8
2994
5017
7027
9024
1009
2980
4939
6886
8820
074a
2653
4$5o
643*
8310
017a
2023
3S62
5^96
7107
93 '2
tio6
2889
4662
6423
««7?
9913
1643
3361
$059
6768
84«
6709
8972
1220
3413
5571
7874
oosa
2236
4395
<S539
8670
0786
3889
4978
7053
9«'4
1162
3£97
$219
72»8
92*3
1206
3«77
7080
9013
0934
i^»
4739
6624
8497
0358
2*07
4046
$872
7688
9492
12S5
3067
4838
6598
8348
0087
1815
3532
5240
<5937
8623
7616
9873
2115
4542
^554
8751
093J
3101
5254
7393
95i»
1629
J726
S«o?
787?
9535
1978
4007
6024
0018
t99d
3961
59»4
7854.
9783
1699
360a
5490
7374
9242
KJ99
2944
477»
6600
8411
02 iT
1999
3777
554J
72951
9045^
0779
»50|
4216
$920
76*3
2225
8970
70.1151
3917
5469
7606
.. 9730
71.1839
393$
<oi7
8085
72.0140
2t8l
4»ID
6225
8227
73-0216
219}
<i09
8048
74.1889
3792
5«g
756?
9429
75.1284
3»28
4960
6781
8591
76.0390
2177
3954
5720
7475
9219
77-0952
2675
4388
6090
7781
^3H Ld^ar.SifUSy Peg. 3lij9jl9%¥>%M%Ai^4i^jUJ^x.<i,
934^
4t49
SS72
Sao4ft7
»?73
5o$8
^557^7^9 68
io67
95^99718
J7^
^477
84*0559
. 4?72
5663
6944
. 821J
4308 4467
8218
J919
S»34 159
9339^ 9495
^34^
686c
98^7
IJ58
2840
4313
5778
7J35
0120
»55o
297a
43?6
^f-f^-J-
8370
9988
•59^
5'9$
4784
^3^
7934
7049^71^9
8578
99SS>
I3J2
1697
41!7j4£^
593Tr^07i
7?2« 7467
8716 ,3855
0097 ;^^H
1469 1606
4189 :432s
^n^ 5672
5878 7oc»
82 M j8344
4[95l6 9^8
0^14 0985
^60 4889
g^}99 8726
djoo id^dr b6u
19^
*5a3
5'a69
85;^
57«»
5433
7059
869.^
OE490310 0471
»T57
3354'
4942
8091
4419
7145
8477
5£a
3198
59^r
5597,
■1917 *«o77
3513
5679
«247
9805
2897
442f^
1595*
74^
8969
04^5
1952
34ra'
4900
^6x
781I
9257
0693
2I2Cr
5$3«
i^i2
'^35 »
774S
9i1t
0^09
1879
«4i
j83^
5417
5995
:856o
99^2 ^0117
I $11 i66f
367?
5259
'6836
840?
4S9$
594 «
7379
-8do9
9gor:>{:999^|.od5i.
1378
.ij^|6ir247
»424 ^^^55
3725 1^5f
5018:5147
75te (71709
asja
88^6
9pJ[»|?9iio
o5^2 ^
2237
5050
^68s
J984
527^
7836
.3204
45Bif 4734
^254
77^
r9265^
•2248^
3725
$193
6652
7546
8875
Oi9dr
2ffi>5
4114
S404
6688'
79^41
9to6i9^^fo»S #
^9?
5^397
3991
552$
87i£
0^2
1819
35<7
'4895
(5405
7917
2396
3872
8103
954 5
^79
^404
3821
'5230
65ii
8<Si3
9407
3784
2152
^h
4865
6209 ^54$
824:7
968>
ri22
154»
<577p
8i6!2
9545
0921
228S
5648
^99
7^79
9007^
0327
rt4$
294?
4243
^9k
f-*g ^ 4<fgwfr, ay.37, $8, 39,40, 41, »2, 43, u77i. 9. *3J .
I ■ •; ' " 1 I !■ I wtmmm^^^m^ i— M— I , I ^
Tl -'• fl ■ 9 I 9 J . i* . A I . O ^ .. i
37.a»7.7tf4
10
no
40'
SO'
to
30
40
_So]
39.0
«o;
20
30J
40
Jao
10
20
JO
40
4>.o
10
201
3o<
40-
50
4*-o
lO
3BO
3D
40
50
43- 0]
10
20
SO
40
10
20
30
40
1^.2363
4980
7594
0204
2810
8010
90«o6o5
3197
5784
«369
91.0951
3529
6104
$677
2.1
381
637^
8940
93-H99
K
4262
9353
p^. 1-896
7577
ooo}
2625
5«4«
78$$
04»S
3070
S67*
8aro
o?64
345$
604$
8672
U09
0963
«934
1503
0634
9196
'7SS
43,' I
61866
94»^
1968
45«7
71*53
9607
2I$0
8886
14671
4691
7231
9760
2306
4842
7376
9909
«44»
4973
7503
.0033
5090
7618
®'45
2672
$199
77W
-i-l-
76^17903
2887
5S«3
8116,
oy«5
335<
595*.
1124
|7»4
6302,
619
9'?^
1760
0890
94S«
2010
45<'7
7121
9673
22Z3
4771
73»7
9862
2404
4945
7485
4023
2j6o
5095
7629
01^2
2694
$226
7756
0286
2814
7871
0398
2925
54f*
J«4
576:
03.77
0986
|59»
61 92
8789
138}
3973
656P
9»44
*7«4
4302
6876
9446
2017
4S8"3
7H7
fS?
4822
7376
99i8
H78
5026,
7S7*
oii6
2659
5199
7739
0277
2813
5349
7i?3
0416
2948
5479
8099
0538
30^7
5J96
8183
q6{i
3178
|7<>S
2211
4_
8165;
0790
3410
6026
8639
1247
3851
6452
9049
<042
6819
940?
1982
4560
7«34
9705
2*74
4840
74*3
9964
2522
5078
763^
0183
2733
S281
78i«
0370
29y
54<3
7995
0530
3067
5602
8136
0^69
|20t
S73«
8^62
P791
31JO!
5848
8376
0993
8428
.Lo;*
3672
6288.
8990
1'"
«7P*
939«
1991
4491
7077
9660
2240
4817
739 J
9962
2ii30
5096
7659
0219
2778
S33S
7887
0438
2988
55JS
0625
3»^7
585!
838s
5707
8246
0784
3|20
_-?
09??
^985
8SfS
1044
3$£3
;6toi
86119
ii$6
_6_
8691
IS 14
3934i
6549
9(60
tzi'i
437i
6971.
9JM:
2i4o:
4750)
7336
99' 8
2498
0219
2787
S3S«
79' S
0475
3033
$589
8t4»
0694
3243
5790
8|35
0879
5961
8soo
1038
3$74
6109
8613]
117s
D238
8768
1297
3826
7
8953
1576
4*96
6810
9421
2028
4632
7«3r
9827
2419
$008
7594
0177
27j6
533«
7995
0476
5609
8'7«
07|i
3?89
844
197
0949
3498
6045
8590
'«13
3675
6215
8754
1291
J827
6362
8896
142?
960
49>
9021
1550
4079
92 1£
1839
4457
7072
9682
22«9
489*
7491
UO86
2679
«?07
7852
0435
joi-4
SS90
8163
0733
3300
J86s
8427
0987
3545
01.00
I20<f
37^2
6299
'^354; 6607
8881^9134
1400^ 160^
^7 68
1387
$929
6469
90 d8
'S45
4081
0616
1682
±213
6744
9274.
1803
433 «
6860
947«)
2IOII
47^9
7333
9943
*549
5152.
77SJ
0346
2938
$526
8iii^
0695,
3Vt
S847
842q
3S57
6X21
8683
1243
3S0Q
6355
99oi
H55
4007
6554
— 164.2
1
1042
4j8^
^7«3
92^2
799
4335
68<59
9403
9.387
*9'4
444»
6968
949St9747
4466
6997
9S*7
2056
4584
96^0
2107
14694
7221
'40 £gf . Sims, Peg. 45, 46, 47, 48, 49; 50, 5 1, 52. In. q
D '
4J.O
to
»o
30
40
so
84.9485
85.0745
>997
3*43
•4480
5711
I
96, »
0870
336^
4603
5^33
7056
8272
9480
0682
1877
3064
4«45
$419
6586
7747
8900
0047
1187
?3*»
3448
4f68
5681
6789
7889
8984
0072
IIJ3
22«8
3297
4360
6466
7510
8549
9S79
605
624
637
111
641
564
529
489
2
9737
0996
2247
3490
47«7
iiss
7178
8395
9601
0802
1996
3183
4363
5536
6703
7862
9015
0161
1301
2434
3560
4679
$793
6900
7999
9093
0180
1261
2336
3404
44^6
5521
6571
7614
8651
9682
707
>26
738
]%
729
712
689
660
626
585
^64
1121
237J
3614
4850
6078
7300
8JI4
9721
0921
2IIS
3JOI
4481
J<53
6819
797«
9130
0276
HH
3672
4791
5904
7010
87 09
9202
0289
1369
«44J
3510
4572
6675
7718
87JS
9785
809
827
839
846
846
840
•lis
810
787
757
722
681
_+_
9990
1246
2496
3738
4973
6»oi
984,
1041
"34
34«9
4598
5770
6935
8094
9*4$
0390
1528
26'i9
3784
4903
6014
7120
82T8
93"
0397
•477
2550
3617
4677
gu
7822
8858
9888
~9"
929
940
946
945
_939
926
908
884
854
818
776
01 72
137*
2620
3862
5096
6323
Z5'3
8756
9962
1 161
*353
3$37
*VJ
5887
7051
8209
9360
0J04
1641
2772
3896
5014
612$
7230
8328
9420
0505
1584
26$7
3723
4783
6885
7926
8961
9990
013
030
046
04$
^38
I2S
006
981
9fi
87*
6_
0242
'497
274$'
3986
$219
644$
8877
ooUj
• 280
2471
3656
4835
6004
7167
8324
9474
0618
«7$S
2885
4008
6236
7340
8437
9529
06.3
1692
2764
3829
4889
5942
6989
8030
9064
0093
"5
132
142
146
»44
-137
123
104
078
'047
010
967
_7_
0367
1622
2869
4109
$342
6568
7786
8998
02O2
'399
2590
IZ74
49$o
6120
7283
8439
9589
07? 2
1868
2997
4120
S237
6347
74$o
8547
9637
0721
1799
2871
3^3«
4994
6047
7693
8iS3
8167
o'95
"217
233
242
246
244
236
222
201
176
144
106
063
C49J
>747
2yP4
42 J 3
$4tf5
^690
7Po8
9119
0322
1519
2709
389a-
50(58
6237
7i99
8$55
9704
084$
1981
Jl'io
4252
$348
<5457
7') 60
8655
9746
0829
1907
«977
4042
5100
6152
7198
8237
9271
0297
3?9
334
343
346
343
_3JS
320
299
273
240
202
«58
9
061^
3118
435<5
5588
6812
f6.o
10
zo
30
40
50
47.0
10
20
30
40
50
48.0
10
20
30
40
50
49.0
10
so
30
40
50
50.0
10
20
30
40
50
JI.O
10
20
30
40
52.0
10
20
30
-5°
6934
8150
86.0562
i75<
2946
4-»7
5302
6470
7^31
8785
87.993 «
1073
2208
.3335
4456
5570
__<5678
7780
8875
«»9963
88. 104$
2121
3191
4254
7406
8444
9476
890.503
891.523
892.536
894-54^
895.542
896.532
897.516
898494
899.467
890.433
901.394
8029
9329
0442
1638
2827
4010
5185
«J53
8670
9818
09*0
2094
3223
4344
54S9
7670
87*5
9855
«9}7
2014
3084
^148
Saos
6257
7302
8341
9J74
0400
402
4J5
444
446
443
J33
418
397
370
298
253
Log. TiiMg. D4g. 45,^, 47» 4*, 49* 50» 5*» S*- J>i' lo. •41
V^J^^^„^M^— I I I » I ' ■■■■■■ II ... ... C . ^^^ . - -— ■ ■ -
20
30
47.0OJ.O344
. 10 a877
5412
ju 7947
40 04.0484
48.0
10
20
30
40
JO
49.0
10
20,
50
40
50.0
10
20
30
40
$0
55<53
S104
05.0647
3191
573»
8285
0608^7
3389
5944
8501
07.1040
^622
61S6
8753
l|23
3895
^471
9049
08.1)23
51.009:1^3
10
20
SO
4215
^«P3
.- 9995
40 to«i99o
'" 45?3
$2.0
10
90
12
7^,
-^ 979^
ftOti«24o6
30 1 1019
4Pl 7^37
joi2.o2;»
2- 1 '?
0535107)58
3oj;a 32B5
39 5811
06 1 ^ oi66
i?4P 3?P3
0597
3131
5^^5
8201
07gB
3*77
5817
«358
0901
344^
J992
8<4i
5^(^8 59*^ ^174
8*97 9450 8703
P726 0979
3255 5509
57J<7 do4o
8318 8571
lie 4
6172
8708
r2.;6
3785
0851
3384
919
^455
0992
3531
1092
3^44
5200
^757
1316
J878
6443
9010
liso
4153
674
9307
1889
4474
7062
9654
2249
4848
7451
0057
766y
5281
7899
0522
6325
1-410
3955
d$02
9051
»357
3891
8962
1500
4738
5071
ii$6
3701
6248
879^
347 *^2
3900 4155
6455 6j ii
9013 9269,
1573 1829
4235 43^9i
6700 ^956
9267 95^4
t837 2094
4410 4568
^986 7244
9565 9823
2147 240^
4733 4991
7321 7580
9913 0173
25C0 2769
510815368
6'>79
9121
1^54
4210
<^757
9305
77*1 797
0318 0579
2928 3189
5543 $804 6066
8161.8423 8686
0784 1047 1309
4
ion
3^37
6064
8591
u;8
3^4tf
8<44
137F
1232
3762
6293
8825
.6427
8956
1485
4015
6546
9078
1858
44»i
6467
9524
2085
464S
7213
97«o
2352
4925
7502
0981
2664
5?50
783?
0432
3028
5628
8232
0839
3451
1611
4144
6680
921"
»753
4290
6833
9375
1919
44^5
7oii
9561
2113
46J6
7222
97^0
2341
4904
7470
C038
2609,
77^0
03 jp
2923
5509
8099
0692
3288
5889
8493
1 100
37ii
6g28
8948
1572
1516
4045
6^69
9096
1624
4x52
6680
9209
1758
4268
6799
9331
1864
4398
6953
9469
2007
454^
7087
9629
2173
4719
)26:
9816
2368
4922
7478
0036
2597
516c
7726
0295
2866
544P
8018
0598
3m
5r^8
8358
0951
3548
6I49
8753
1351
3974
6590
9210
1835
7
1769
4295 r.^
^8*? 7075
9349 9^<52
1877 2129
4404 4^57
6932 7185
9461 97H
1991 2?44
4521 4774
7052 7J05
9584 2!iZ
2117 *37*
4551 4905
71^7 7440
9723 9977
2261 2515
4«oo 5f 54
7341 7595
9884 <>>38
2429 2682
4974 5229
7522 7777
O072 ^27
2623 2879
5177 5433
7734- 7989
02p^ 0548
2853 3110
5417 5673
7983 8246
0552 0809
3123 3381
569S 5956
«275 8533
0856 nr4
7440 3698
6026 6286
86i7|8876
1211 1470
3808 4068
6409 j 6670
907919275
J—
2021
4548
1622
4235
6852
,9472
2097
1883
4496
7i«3
9735
2360
2-74
4861
7328
9 55
2382
491£
7438
9967
2497
5027
7559
0091
262^
5158
7694
0231
2769
530*
7850
0392
2937
5483
8032
0582
3»34
C689
8245
0804
3366
5930
8497
1066
3638
6213
8791
1171
3957
6544
9135
1730
4328
6930
9535
2145
47s 8
7375
9997
2623
• F
fc.
l%m*
m-
• yy»
t-
7
8
OJ4
loS
9<,9
059
V98
99 K
S32
9-5
76.
^5^
68a
7_74
S99
690
510
601
'IM
sotf
3^^
405
210
ay9
099
187
pS2
070
860
948
733
820
6co
6S7
462
^48
318
404
It9,i54|
C15
099
K55
P?9
65)1
774
^20
603
34^
427
164
H^
97^
060
787
858
55*
<57i
^90
469
ii?
262
972
oso
7^5
^3^
'i??
6n
5:^6
383
074
Ml
538
914
59^
671
349
4H
(^97
171
F40
914
57^
652
511
Ll5
040
113
763
S5I
463
554
^96
2^7
^o5;975|
60?
<579l
781
691
495
388
158
©35
907
4S9
33S^
184
02 g
857
685
509
328
HI
949
751
549
i±!
129
911
688
460
990
747
499
Z46
988;
725
455
185 ^
908
625 ;
938
046
749/
£^, TangiHts^ Peg. 53, 54, 55, 56, 57,^ 58, 59, 60/ fn.
10.
D '
53.0
10
2a
30
JO
10
20
SO
40
50
55-0
10
2Q
30
40
Jf>
10
ao
30
4^
52
57-0
10
20,
30
40
50
58.0
10
20
30
.40
—12
59:0
ip
20
10
40
60.0
lb
,ao
30
40
50
{Z.28S6
5516
8151
13.07^1
343^
/6o8$
-8739
14.1398
4062
6732
9407
15.2087
4773
7 16$
164O162
2866
5575
8291
17,023
3741
^476
9217
18.1965
4720
74^3
19.0252
3029
. 5813
8604
20.1403
. 42n
7026
9849
21*2681
5521
.8369
22,1226
4092
6967
9^51
23-^745
564S
8561
24.1483
44«S
7358
25.03U
3274
3148
5780
8415
1055
3700
9005
1664
4329
6999
2356
5042
7734
043^
3136
5846
85^3
120$
4014
^749
2240
4996
7759
0529
33^7
6091
8884
1684
4492
7308
0132
1964
5805
15x2
4J79
7255
0140
303 s
5939
8852
1776
4709
7<^S3
0607
357'
341 1
6043
8679
1310
3965
6615
9270
1931
459<5
7266
9942
2624
8004
0702
3407
6ii»
8^
I158
4287
7023
9766
2546
5^72
8036
0807
3585
6370
9163
1964
4773
7590
0415
3248
6q9o
8940
1799
4667
7543
0429
3325
6230
9144
2069
5003
7948
0903
3868
3674
6306
8942
^584
42 JO
^880
953^
2197
4863
7534
0210
2892
t^
8273
°973
3678
6389
9107
^830
4561
7297
0041
2791
5548
^313
1084
3863
6649
9443
224$
5054
7872
0698
3532
^374
9225
2085
4954
783*
0719
3<^i5
6520
943^
2362
5297
8243
1 199
4165
-^
— 1.
3937
6570
9207
1848
4495
7146
9802
2463
513^
7801
047S
3161
5849
&543
1243
3949
6661
$379
210^
4834
7571
0316
3066
5S24
8589
1362
4141
9723
553<5
8154
0981
3816
6659
9511
2371
524)1
8129
ipo8
3905
6812
5_
4260
6833
947*
2113
4760
7411
9728
2655
5592
853«
|H95
F7
0068
2730
5397
8069
0746
3429
6ii8
8813
»5i3
4220
6932
965J
^37^
5107
7S45
0590
334*
6i_oi
8866
1639
4420
7207
P003
i8o6
5617
853^
1^64
4100
6944
?7y7
2658
5529
8408
1297
4195
7103
P02I
2948
5885
8833
1791
14760
6_
4463
7097
9735
2377
5024
7677
0334
2996
5663
833<5
1014
369J
6387
9083
£783
4491
7204
^23
2649
5381
8iio
0865
3618
fill
9143
1917
4698
7487
0283
3087
5899
8719
1547
4384
7229
0082
2945
5816
8697
1586
4486
7.394
0313
324X
6180
9128
20.87
5057
4727
7360
9999
2642
5289
7942
0600
3263
5931
8604
1283
3967
6657
9352
2054
4762
7475
0195
2922
5651
8394
1 1 40
3893
6653
9420
2195
4977
7766
0563
3^18
6180
9001
836
4668
75 H
036^8
3231
6104
8985
1876
4776
7685
0605
3535
6474
9424
^384
5355
4990
7624
0263
29q6
5555
8208
0866
35*9
6i9«
8871
1551
4236
6926
9622
2324
5033
7747
0468
319$
592B
8668
'415
4169
6936
9697
2473
5255
8045
084J
3649
6462
9284
£114
4952
7799
0654
35i8
6392
9274
2166
5067
7977
_f3
^_
5253
7888
0527
3171
5820
1132
3796
6465
9139
1819
4504
7195
9892
^595
5304
8019
0740
3468
0202
8943
1690
4445
7206
9975
275*
5534
8325
1123
3930
<^744
9566
2397
5236
80^4
0940
3805
6079
9563
M55
5357
8268
o8?8
3828
6769
9719
2681
1190
4122
7063
0015
2977
5950
^/^^Log. Sines D^j^. 6l, 62^ 63, 64, 65, 66, 67,
^■An.i^i
D »
I
£
I
JL
.♦
d
U 8
i-
Si.
941.819
889
959
029
699
««>
ftj9
'^48
10
942.517
587
6^6
7*5
7£5
854
P3J
003
o?2
141
20
943.201
279
34^
4»7
466
155
5i4
6fi
76.
Sjo
30
898
967
036
104
I7i
441
309
%7l
44<
5'4
40
944-582
650
7*8
7|6
136
«54
92t
990
0,8
125
8?|i
538
50
62.
945.261
328
002
596
069
53»
203
SP8
270
686
337
733
+04
81J0
471
935
10
94^.604
671
73«
804
870
937
004
070
tjd
203
i
20
947.^69
335
4ot
467
533
599
665
■731
797
863
30
929
995
c6o
12^
192
257
3«3
388
4^3
519
40
948.5^4
649
715
780
845
910
975
040
165
i7er
^52
94^?35
300
3:.4
419
494
558
623
(588
7';»
U6
<53.
881
945
009
C74
138
'202
•266
330
^94
428
10
950,522
58d
650
7iif
777
8+t
9d5
96%
032
096
20
95*. 159
212
286
34:9
412
47«
.539
604
6bs
728
30
. 791'
854
917
980
043
JOS
168
2Jt
294
35*
'40
952.419
481
')4+
606
66%
731
79J
8js
917
979
$c
9S3-042
104
166
228
290
351
413
1^
537
59«
64. c
660
722
78?
845
9CS6
96T
029
090
152
215
.
10
954.174
335
39*
457
J18
',79
^40
701
762
82)
2C
883
9*4
005
065
126
i«i5
247
307
368
428
. 30
y55.48S
'^!
609
669
729
789
849
909
969
(}2a
4C
956.089
148
»o8
268
328
387
447
506
<,66
d2,
50
65.
684
957-276
744
•335.
804
353
862
452
921
980
570
040
<5i^
oJ>9
•687
158
746
217
3(^4
10
862
921
979
038
096
»54
212
271
329
38^
3«s
29
95S.4;$
5^5
561
619
677
73+
792
850
^8
3c
959.023
080
>38
195
253
31b
3«7
4*5
482
539
40
59^
6-:i
711
768
%^S
881
938
99$
052
109
66.
960.16$
222
786
279
84s
335
899
392
95)
44»
oil
i?5
067
5*t
12}
($18
179
674
i35
730
10
961.290
34'i
402
45^
5»3
5^9
,624
(J80
^35
791
* -20
846
902
957
012
067
123
>7«
^33
288
345
»
30
962.398U53
50b
562
<5i7
672
727
781
836
891
40
94< 999
054
io8
I(J2
217
ift
32s
379
434
in
50
963.468
542
59f
650
704
757
»ii
86,
919
67.
95, o:6
080
'33
l8"7
24D
294
347
400
454
S07
10
56O
6'3
66<J
719
773
»2(5
878
931
984
073
20
965 .090
143
i95
248
301
353^
'40*
458
Sii
5rf^
30
6i5
(568
720
772
8^4
8^9
•928
980
052
dSf
40
^66.136
188
24<J
292
344
?95
447
499
556
60a
SO
653
705
75!^
807
SS8
910
961
012
064
i£5
68. 9^7*166 217
268
319
3(59
420
471
522
573
^23-
/
10 674 725 *
775
826
87tf
927
977
0J7
o?8
ul
' zo 968.178 228 !
78
528
?79
4i9
478
528
57&
628
30 678 728 )
^77
827
877
p36
97^
02J
075 r24
40, 9^9.173 229J>J
.72
121
370
420
♦69 1
5«8
5^7 616
l^ 501 665*714^762 \\
Jii 86o|909J<
?57'oo6|<
6ilo «J^5toJ5 61545
>o S0.ae29 25129
5255 5557
}6|, fat, ^63» 64» 65, ^6» ^v^lftj i5i> fo. 45J 1
b<5
40 6254. 8556
5P ?Z:1£^ il""
30
50
10
.30
40
6(5. b
16
20
36'
40
_J0
67.0
10
20
40
ill ■.,
62.0 ' 432^'^ 4^30
10 7379 7^85
bo it*o445 ^752
• 3523 J«32
6614 6934
9718 0019
6^"^ 115^1834 3146
5964 6277
9i«7 942a
30,22^4 2580
5434^ S752
^50 _.86i9 8938
64.© jt:i8i« 2139
10 5<^2 5354
20 8260 8584
30 }2.rjo4 1829
40 4763 5CJ89
_-l5 ^ o?;^ ?565
65.0 j$it327 1657
10 4^34 4965
i201 7>95!7 ^290
30 J4.1296 103*
46 46ji 4989
•56' 8t>26.j8364
35:1417 *7S7
4826 i 168
^2*5 3 ^596
35.i'698 2^44
5t6t 5510
_»645 5?f
J7.2148 2499
5671 6024
9213 9568
30 36.2776 3133
6359 67'?
^964 0326
68. D J6.3590 3954
i6| 7239 7605
20 40.0909 1277
40
<6
46044973
8J19 8692
^f.^ogp 2434
.6844
2829
583^
8859
1891
493S
7991
1060
4140
7234
0340
3459
6591
9737
2897
'fcyo
9*58
2460
5676
890S
2154
5416
8694
1987
^297
8623
1966
.53*6
«702
209^
5510
8940
23S9
5857
9344
6377
9924
349«
,707^
0689
4J»8
1646
5344
9055
2809
7142
10130
3*29
6140'
9162
524.0
8298
1367
444Q
7544
0651
377^
6905
0053
3^13
638S
9577
2781
5999
9292
24Sb'
5743
9023
2318.
5629
8957
2301
56-^2
9041
2438
5852
9284
6205
9694
1203
0279
3849
7438
i?5^
mi
*0|{
5?i$
94J8
3185
4-
40^4
7219
0368
3530
6707
9897
2648
5961
9290
2636
6000
9380
2.7T8
6194
9629
3081
6553
OP44
7555
7o8j
0635
4207
7799
1412
$047
8704
2384
ao85
9812
ij6i
5851
891 1
1983
$067
8164
1.274
4397
0684
3847
7025-
0217
664)f
9880 t)2l>5
^39«
9680
2979
6293
9624
2272
6337
97x9'
311*9
6537
9973
3428
6901
0394
3907
7439
0992
45^5
8i<9
izis
J4I2
9371
.2753
645J
0186
^938
6155
9217
2291
5576
8475
1586
477o
7848
0999
4164
7325
05371
3745'
6967
S457
6725
D009
6645
9958
33o«
oo;8
^60
6880
o3i8i
3774
7250
0745
4259
7793
1348
492?
8519
2137
5023
8163
i3?5
4482
7662
08^7
4c^66
729'>
05^9
3783
705}
OJ39
3640
6958
0292
36H
7012
039^
3801
72i5
0662
4121
7598
4612
^148
1705
^528)2
8880
2500.
J777
9438
6829
0560 {0934
^^Lagar.Sintt.Dtg. 69, 70, 71, 72, 73, 74, 75, 76. 7i». 9^
1> •
69.0
10
so
30
40
50
70.0
10
>o
30
40
50
71.0
10
20
30
40
.?°
71.0
10
«o
30
40
73.0
10
20
30
40
50
74.0
10
zo
, 30
40
75
to
*o
30
40
$0
76.0
10
20
30
■■40
$0
1
soo
683
t6i
635
»05
570
032
489
94*
391
836
*77
:u
574
999
!»
«47
65$
059
247
635
019
399
11+
878
238
594
945
293
637
978
646
974
299
61.9
936
248
III
163
460
2
248
7'i
208
682
6ir
078
987
881
321
757
8J7
288
696
100
g?
286
673
057
436
812
183
SSI
914
629
980
328
672
on
347
679
007
6ji
9-6-7
279
587
892
>93
490
3
297
779
256
729
198
662
*5+
580
032
4i«i
93s
3^5
800
232
660
083
50J
?l!
329
73<5
140
539
934
325
712
095
174
849
220
587
950
IS
706
380
712
039
683
998
310
618
9*2
222
SI9
4
IS
r4
-245
709
625
077
S2J
969
408
844
275
702
•25
544
959
370
777
180
579
973
364
750
«S3
512
836
62J
986
as.
7001
650
3*97
7J0
079
414
745
072
395
7H
030
341
649
9SJ
252
548
X
874
1^'
823
291
75 S
671
122
570
013
412
887
3«8
167
586
001^
4n
817
220
618
on
403
789
»7»
549
924
294
6&)
022
380
085
43«
Z.74
112
778
104
061
272
679
983
282
578
6
442
922
398
870
338
802
zS'i
614
057
496
930
361
787
209
628
042
til
260
658
05*
441
8*7
209
058
416
770
120
46<S
808
146
480
811
»37
459
778
092
403
710
913
do7
-1
490
970
446
917
384
848
761
212
659
lot
539
974
404
830
25 »
669
•83
493
898
300
-697
091
480
866
247
«24
X
733
094
A51
805
»5S
500
843
180
513
843
169
491
809
124
434
749
043
Itl
8
538
018
493
964
1''
!2+
352
807
257
703
145
5ii
017
446
872
"93
7"
1*4
533
959
340
737
130
5«9
904
??S
662
035
4«4
no
i
8jl
213
547
876
202
5*3
841
465
771
073
9
^^
065
540
on
477
940
398
852
302
Ig
626
o<5o
489
914
335
■M
$74
979
380
169
558
f«
699
072
805
166
875
224
569
910
247
580
909
234
V^
21
lis
801
803
n\
97o.i$2
«3J
971.H3
97«-os8
524
986
973443
897
974-347
792
975 233
670
976.103
532
957
977-377
794
978.206
6»s
979019
. 4*0
8f6
980.208
596
980
981.361
737
982.109
477
" 842
983.202
SS8
910
984-259
603
985.280
613
945
986.2<56
587
904
987.217
526
988.133
430
Ltgar. tang. Dig. €^^^6,^1^ jt^fi, *j^ 75, 76. tn. 10. \y
69*0
10
20
30
40
70.0
10
20
30
40
50
71.0
to
20
30
40
50
72.0
10
20
30
40
J2
73^o
10
20
30
40
J?
74.0
10
20
30
40
JO
75.0
10
20
30
40
50
76,0
10
20
30
40'
50
54.2504
7294
55.2130
70IZ
56.1941
6920
57.1947
7026
58.2157
7342
59.25S1
7876
60.3229
8640
61.4112
9646
62.5244
63.0906!
6200
999^
3807
7648
15 14
5407
9327
3274
7250
1253
5285
9347
3439
7S6i
1715
5900
0118
4369
8654
2973
73*8
1718
6144
0610
5^13
9655
4237
8859
3523
8230
2981
7775
2616
7502
2437
7420
2453
7537
2673
7«63
3108
8409
3767
9185
4663
0203
5807
1476
6578
0371
4190
8033
1902
11?!
97*1
3671
7649
1655
5690
975J5
3849
7975
2I3Z
6320
0542
4796
9084
3407
7765
2159
6590
5565
am
4697
9324
3992
8703
3458
8257
3102
7994
2933
7921
2959
8048
3190
8385
3636
^42
4306
9730
5214
0751
6371
2047
.3
6956
0752
4573
8419
2291
6189
0115
4067
8048
2057
6095
0163
4261
8389
2549
6741
0966
5223
9515
3841
8203
2601
7035
1508
6018
0568
5158
9789
4461
9177
3936
8740
3589
8485
3430
8423
3466
8560
3707
8908
41(54
947f
4846
0276
5766
1319
693 s
2618
*- -V
7335|77i4
4956 5340
8805 9191
2679 1 3068
658i|6972
0509 0903
44641486.
8448 18847
2459:2852
65t)i ;69o5
0571 '
4672
8804
2967
7162
1390
S65£
9946
4276
8641
3043
7481
ii57
6471
1025
5619
0254
4931
96$i
4414
9223
4077
8978
3927
8925
3973
9072
4225
943 X
4692
0010
538'6
0822
6318
1877
7501
3190
0980
508"4
9219
3385
7583
1814
6079
0378
4711
9080
3485
7927
240J7
6925
1483
6081
0720
54CI
0125
4893
9706
4565
9471
4424
9427
4481
9585
4743
9955
5222
§145
59^7
1269
6871
2434
8067
3762
8C93
1896
5724
9578
3458
73^
1197
5258
9248
3265
7312
ii?£
5496
9634
3803
8005
2239
6507
0809
5146
9519
^917
8373
2852
7379
1 94 1
6543
1186
5871
0600
5372
0190
5055
9964
49-2
9930
5908
0049
4222
8427
2665
6936
4989
0099
5262
0479
5751
1081
6^69
1916
7425
2997
863^
43^6
7_
8472
2277
6108
9964
3847
1692
5656
9648
3669
7719
1798
1241
5582
9958
4370
8820
11^
7833
2399
7005
i6.j3
6342
1075
5852
C674
5542
0457
5421
043A
5497
0613
5781
1003
6282
1617
7011
2464
7979
3558
9201
4910
J [ 9
885119231
265913041
6493
03S2
4237
8148
2087
6054
0049
f072
125
2208
6321
0465
4641
8849
3090
7365
1674
6018
0397
4813
9267
3^58
8288
2858
7468
2120
6814
Vl5i
6332
6031
0951
5920
o9j8
6oc6
1x27
6301
S29
6813
2154
7553
3013
8534
411:9
6877
07>9
4627
8541
2483
6452
0450
4476
8532
2618
6734
0S81
5060
9272
3516
7794
2107
6454
0837
5257
9714
4209
8743
33*7
793*
2587
72S5
2027
5813
1644
6521
1446
6419
M42
6515
1642
6821
2C55
7344
2691
8097
3562
9090
4681
9768|C337
5484' 6060
ir
I
i:
loz^r Tang.Beg. 77, 78, 79, 80, 8 J, H2, %i,M ^ex rTTmr
77- o
10
20
30
40
_5o
78. o
10
20
30
40
-1?
79. o
10
20
30
40
80. o
10
20
30
40
JO
8i7o
10
30
30
40
50
82- o
10
36
40
JO
83- o
JO
20
30
40
50
10
20
30
40
.-JO
• 64.2434
• «303
^54245
iJ6.026i
6354
•67.2525
.0^778
68.5115
69. 1 $37
8049
70-4651
71-I348
8 141
72.$03d
73-2035
9»37
74.6352
75.3681
76-1128
86p8
77.6393
78.4220
79.2183
8o.o2d>
8538
81.6940
82.5501
83.4226
84.3123
85.2197
8^.1458
87.0913
88.0571
89.0441
90.0532
91.0856
92.1424
93 .2*4^
94.3340
95.47 »6
95.5391
97-8380
99.0702
11-00.5376
01.6423
02.9867
04.3733
7213
3018
^894
4843
0867
6967
3 '47
94c8
$753
2184
8705
$316
2023
^826
5731
2739
98^4
7080.
4421
1880
9461
7170
50H
2987
1 106
9371
7789:
6366
$108
4022
3115
2395
1870
1548
1440
1^4
1902
2495
3345
4465
5870
7575
9597
953
4663
7749
1234
5144
7790
3^2
9486
5442
1473
7$82
5161
2634
Ot26
7948
5802
3793
1926
0206
8640
7233
5992
4923
4034
3333
2828
2528
2441
2578
2950
5567
4444
5593
7027
8763
0817
3208
5955
9079
26c6
6560
418
0078
6042
2o8t
8197
4393
0670
7032
3481
0020
6650
3375
0199
71^4
4153
1290
8539
$9>^3
3386
0993
8728
6595
4600
2747
1042
9492
81C1
6877
582^
49 $6
4274
3789
3509
3444
3604
40C0
4644
5547
6723
8187
99S3
2041
4466
7250
0413
3981
7979
8947.
4773
<:i67i
66^2
26^9
8813
5017
1303
7673
4^31
0678
73f8
405"3l.
08S7
7822
4862
2016
|27o
6646
4141
1761
9$o8
7389
54c8
3570
ibSo
0345
8971
7764
^!?
5879
4751
4493
4450
i^ii
$053
$722
6652
7856
9349
1 148
3267
$728
8549
1752
5361 16745
5403 10832
5220
3561
2057
0716
9543
8546
7167
7887
8870
0131
1684
354s
6894
.3205
9601
6086
2660
93»9
4071
1858
9780
7841
6047
4403
29U
1591
0435
9456
8660
8056
7652
74$7
7481.
7734
8227
8973
99»4
1273
2856
8
ii
I 9
I269!ib5i
7124,7713
JO-) I 3647
90$2 9556
$1295741
1285,1905
752i|8i49
3841:4477
02460891
67397393
3323,39^7
0001)0^74
677$!74$8
36494542
06251329
770884*2
4900I5626
2205:2943.
96290378
6414. 71747935
4844:5618
2644I3432
05^013^1
86j$'947i
08767706
$i47|6o93
37764638
2468
1329
6368
9591
9006
862?
84499444
84959513
87729813
92900355
062
ucc
1418
4031
4749; 5 95 6 7166
6959182109454
•oSr
9535
2468
5791
9527
3705
^.78
714s
0925
5148
3346
2225
1282
0524
9959
959^
1153
2219
3565
5209
2192
5101
8504
^2<5
6596
;*: -■*■• ,
:§:
p^
-.■';>-w — y^
Mijuj 8s Peg. I gegPeg.
Tbi Remaindgr of the Logarithn Tangents to 90. ♦c;
o
t
3
4
J-
6
7
S
9
•12-
II
la
16
t?
18
t9
2t>
21
21
H
il
15
27
t8
29
12.
31
32
33
S4
36
57
3«
59
40
11.05804a
9506
1.0^0958
u 243$
I- 3907
I. 5584
I. 6856
I. 8j5j
u 9845
K071344
I. 2844
.»• 4351
K 5864
I. 7381
I. 8904
i>o8o43^
I. 1966
I. 3505
I, 5-049
I, 6599
I. 8154
I. 9715
1.091281
I. 2«53
I* 4430
u 6013
I. 7602
I. 9197
1.100797
I. 2404
I. 4016
I. 5<^34
I. 7258
I. 88?8
1*110524
\. 2 167
1. 3815
I. .5470
I. 7131
I. 8798
1.120471
l.i«35«
I- 7«75
I. 9092
1,160837
I. 2<579
»' ^45^9
f. 6387
I. 8152
I>.i70i26
1% 2008
I. g897
I- ^79$
I* 7702
I* 9616
1. 181539
u 3471
I- 5410
!• 7359
I- 9317
1.191289
I. 9258
I. 5241
»• 7235
1. 9236
1.201248
I. 3269
I. 5299
I. 7338
!• 9387
1.21 1446
'♦ 3 5H
!• 5592
I. 7580
'• 9778
I.22I886
I. 4005
I. 6iJ3
!• 8272
i. 230422
I. 2582
I- 47'5ji
87 Deg.
• «53
. 7861
•30C383
2919
5^71
8037
1.310618
3ii6
5828
8456
a8 Deg.
40519
44930
493^7
53890
J58440
6307&
67685
7238*
77131
81932
, 86787
. 91696
, 96662
,601585
. 06755
1 1908
17M1
22378
27708
33105
758078
<^5379
7*80$
80359
88047
95874
803844
1 1964
20237
28672
37273
o
f
' 2
3
4
J^
6
I
9
10
4^48
55004
64149
73490
83037
92797
902783
13003
23469
34»94
45 191
5<5473
68055
7995 S
92191
004781
17746
311U
44900
59142
73855
89105
10490 1
21292
3832^
56056
74540
93^45
114049
2. 35239
^G %
'52
4*
4^
43
44
«-
47
49
50
51
5*
53
14
5JL
5<5
57.
5^
\60
^i Rtmaindtr of the Logarithmit Tttngents^ to go Degrees.
II* 3S9a
It. 5531
ll- 7250
II. 893^
11.130649
II. 2368
IE. 4094
il. 5827
II. 75^7
U.i23«935
1 1 . 9128
11.241332
«'. 3547
"• $773
III. 9L314
11.141068
lu '2839
»i- 4597
II. 6371
II. 8i$4
II. 9943
u.i^i74:>
*»• ^545
86 ENeg.
II. Boil
11.2501^0
II. 2521
". 4793
II. 7078
XI- 9374
11.261683
ri. 4004
If. 63^7
II. 8683
I1.271041
ri. 3412
II* $796
in. 8194
IH.Z80604
a7Deg
11.39302*
II. 6161
II- 9323
c 1.402508
t i' 5 7'7
II. 8949
11.412205
II. 5486
II. 879a
11.422123
II. 5480
II. 8863
11-432272
II. ^709
II. 917^
11.442664
II. 6183
II. 9732
11.455309
II. 6916
88JDteg.
.638570
. 44105
. 49713
• 55379
. 61144
76697-5
. 72886
. 7887S
. »4954
. 97366
.703708
. 10144
. 16677
. 25309
. 30044
. 36885
.-43835
. 50898
158078
89Dcg^
12.2^7516
1 2. 80997
12.30^811
I2. 321^1
12. 60180
»2. 9O143
I2.4223&8
12. 57091
li. 94^80
12.536273
K2. 82030
12.653183
12. 9>i75
12.758122
12;^7304
12.934214
13.059153
i3-23')244
13-536274
Min.
4i
42
45
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Mr S T R E E Ts
T A B L E
^.
LogijHcat Logarithms
To
»/i Day ti PubKJbed^
Neatly printed, in 4 Vols. 12m. Price bound in Cal^. u t.
EmbeUiihed with near lOo Cdpperuplttei, curioufly engraTel
by the bdk Mailers.
Tranflatcd from the t^tnchf by John Kelly, Efqj of the ImuP"
Temple \ Dr Bellanrf of St John's College^ Oiffirdi and J.
Sparrow, Surgeon anci Mathematician.
NATURE DELINEATED : Being Philofophkal
Convcrfations, wherein the wonderful Works of ProvU
dence, in the Animal, Vegetable and Mineral Creation are hid
open, the Solar and PlaneUiy Syftcm, and whatever is curious
in the Methematicks explained. The Whok bcii^ a Compleat
Courfe of Natural and Experimental Philofophy, calculated for
the Inftruflion of Youth, in order to prepare them for an early
Knowledge of Natural Hiftory, and create in their Minds an cx-
akcd Idea of the Wifdom of the G » E at C r b Xt o »•
Written by way of Dialogue to tender the Conception more
familiar and eafy. With a partkularTaUe of Contents tacftch
Volume,
Imulon: Printed for J. Hodget, at AtLnihg-GUfi on landon^Bridge^
Where tikewife may be had, Juji Tuhlijhed^
Dedieatid andprefentei to the Hon, Society of the Imn ftmpU^
(Price bound Two Shillings and Six Pence.)
A New Treatife of Husbandry, Gardening, and other cu«
rious Matters relating to Country Airain : Containing^
A Plain and Pradical Meth^ of Improving all Sort of Meadow^
Pafture, and Arable Land, C^r. And making them produce
greater Crops of all Kinds, and at much lefs than the prefent£x«
pence. Under the following Heads : I. Of Wheat, Rye, Oats^
Barley* Peafe, Beans, and all other Sorts bf Grain. Il^ Tur*
nips, (Jarrots, Buckwheat, Clover, Hemp, : Rape, Flax and
Colefeed, i^c. \IV Weld ox Would, Woad or Wade, Mad-
der Saffron, c^c. IV. Meadow, Pafture Grounds, and the dif-
ferent Manner of Feeding Cattle and making other Improve-
ments agreeable to the Soil of the feveral Counties in Great
Britain. V. Hops, Foreft and Fruit Trees, Vine and Garden
Plants of all Sorts. VI. All Kinds of Flowers, Shrubs in gene-
ral, and Creenhoufe Plants. VII. A Curious Scheme of a Farm,
the Annual Expenceof it, and its Produce. With manyNewt
Ufeful, and Curious Improvements, never before publiflied.
The Whole founded upon many Years Experience. By Samuel
trowel, Gent. To which are added. Several Letters to Mr
Thomas Liveings, concerning his Compound Manure for Land^
with (bme Pra^calObfervations thereon;
^56 ^itt^sUgiJI.LBg0rithms^r\^%^ i
J, 10 1 1, 12, 13, 14, 15. f
8.0
I
74*
V.3-
J-
71J
5
706
6-.
^97
7
688
_8_
679
9
670
1
480
8. 751
733
lO
661
6«
^43
626
617
608
599
59'
582
490
to
$73
565
556
547
539
530
52i
5<3
504
49<5
500
30
487
479
470
46t
453
^5
437
428
420
411
^
$10
40
405
3s>5
386
378
^^
361
353
345
337
328
510
50
JL-I'"
11*
304
288
279
27j
263
«55
247
530
9.
82. 39
3»
23
3'!
58
81
06
07
99
9'
83
75
*7
540
10
V' S9
5*
44
aS
20
12
04
97
89
55®
20
80. 81
73
66
50
43
35
*7
20
12
$60
30
04
97
^9
74
66
59
51
k44
36
570
40
79. 29
21
H
99
91
«4
77
69
62
580
50
78- 55
^7
40
32
-£5
_i8
II
_£}
9^
89
590
10.
77. 8»
74
671
60
■5'
45
38
31
24
17
600
ID
10
oj
96
88
81
74
67
60
53
46
610
20
76. 39
M
25
18
11
04
97
90
83
77
6zm
30
75. 70
63
5<5
49
42
P
28
32
15
08
650
40
01
94
' 88
bi
74
67
61
54
47
41
640
50
74« 34
■a?
21
«4
SI
01
94
87
81
J4
dso
II.
73., 68
61
"54
48
41
35
28
r2
15
<»9
660
10
02
96
89
83
7<5
70
64
57
51
44
670
20
72. 38
H
»5
'9
12
06
00
93
87
81
680
30
71- 75
68
61
56
49
5'
47
.3«
24
18
6j^
40
IZ
06
00
93
"^
81
75
69
63
57
700
SO
12.0
70. 50
44
"84
38
"75
32
7*
26
66
20
60
54
08
48
02
42
96
36
710
720
69. 90
10
30
84
18
12
06
00
94
88
82
77
7$o
20
68. 71
6J
59
53
47
4«
3tf
30
24
18
740
30
13
07
01
'2
89
84
78
7*
66
61
7So
40
^7- 55
49
43
38
32
26
21
«5
09
04
74o
SO
tf5. 96
pi
Jl
81
76
J2.
64
59
Jl
J8
770
13-0
. t*
87
76
25
»o
14
09
03
98
9*
780
10
65. 87
Xi
12
«5
59
54
48
4^
38
790
20
. 3a
27
2t
|6
10
01
00
94
89
84
8OQ
- 30
«4.78
73
67
62
57
S'
46
4»
35-
30
8IQ.
40
2$
20
>4
62
09
04
^1
93
88
83
77
8iO
50
(5g. 72
£z
_57
JU
46
41
36
il
3
830
14.0
lO
<s
10
OS
00
94
89
84
79
74
840
8<|o
S60
10
62. 69
<4
59
54
48
43
38
33
28
23
20
18
1'
oS
03
98
93
88
83
7»
73
30
61. 68
63
58
S3
48
4}
3«
3?
28
23
870
8S0
40
18
»3
08
03
99
94
«9
84
79
74
50
60. 69
<4
^
_55
JO
_45
Jf
35
JO
25
?22
15.0
21
|tf
II
06
01
97
92
87
82
77
900
10
59. 7 J
68
6}
58
54
49
44
39
35
30
910
20
25
20
U
n
06
o4
97
9»
88
«3
9«o
30 S»- 78
74
69
64
60
5S
50
4<5
4»
3^
950
^"^ 1!
27
2}
18
»3
09
1
00
9 J
90
^QAO
so 57- 8^1 8M 77 1 7i[ 68 1 65I
54
49
45 950J
Stxc^'sL^i/f. LcgarAjh ^6, 17, 18, 19, ad, ai,^a ,23. •s;]
1 II
* ^ I J.
j*_
5
6
X
8^
9_
«
16.
i7-4^ 3^ *n»7
22
.1
Ji
P
?♦
00
l6i
10
56.9s
9*
86
8t
77
73
h
64
60
SJ
970
to
5"
46
4«
37
|3
59
!♦
2P
»S
II
980
30
07
ox
98
94
«5
80
76
^2
67
99®
4P
55-^3
5?
(4
S®
♦i
37
33
28
2*
1000
SO.
ao
itf
II
07
03
98
n
90
86
81
iSlf
17*
$4^77
73
*?
«4
60
5«
5*
47
43
39
K>20
10
1^
3«
26
22
18
»4
^
55
01
97
HO3O
20
$3-93
89
»4
8«
76
7a
68
64
5?
55
IO4O
40
5K
47
43
II
35
3«
26
22
18
>4
1050
K?
06
03
94
90
85
81
77
73
1060
52
$2,69
!i
6t
17
53
49
45-
1'
37
33
1070
I& ^
*9
*i
21
»?
M
^
fs
01
97
93
1080
10
SI.S9
85
8t
77
73
69
65
61
57
53
1090
to
40
10
• .49
10
oi
4i
02
P
33
94
29
90
II
22
8z
18
79
14
75
iioo
'1 1 10
50.71
67
6i
59
55
5«
48
44
40
36
1120
49*94
90
as
83
«3
17
79
13
7?
09
7«
05
67,
02
98
60
1130
1140
19* ^
to
^ Stf
s*
♦9
45
4»
37
'1
30
26
22
1150
*5
40
ag^
Is
15
II
0;
U
06
96
92
89
8S
iibo
48.91
77
74
70
63
59
55
Sa
48
1170
4»
37
33
30
26
U
«9
«5
II
1180
04
68
0©
93
97
60
«4
89
21
57
21
85
l9
53
1^
§6
50
55
82
46
lo
75
4a
07
71
7$
39
03
68
1190
1209
1210
1220
4*>
57
53
5«
46
43
39
36
32
1230
30
«9
45.9f
. 5?
as
44.91
25
22
18
>5
II
08
04
01
97
.1240
40
if.
90
56
5«
!4
49
80
46
77
4a
39
70
35
66
32
b3
28
1250
I26q
22
18
15
II
08
05
01
98
94
1270
lO
30
8S
«4
81
77
74
7»
67
64
bo
12ii9
57
24
if
S4
ao
50
47
44
10
40
07
37
04
3+
00
30
97
27
94
1290
1300
40
50
54
|4
Ji
80
47
77
44
4»
70
38
67
34
31
60
1310
1320
zz*
xo
25
21
18
15
II
03
°5
02
98
P
»330
20
*^n
89
«S
«2
79
76
74
69
66
63
134©
to
56
53
50
47
44
40
37
34
31
1359
40
so
»i
a4
21
18
15
12
0^
«>$
02
99
«36o
41.96
' 64
93
tfi
89
58
86
S5
83
5a
80
49
77
45
74
4a
7"
5t
by
36
«37<>
1380
2}.
10
33
30
3
»4
2P
«7
'4
II
Q8
05
1390
ao
02
^
9»
89
86
H
80
77
74
1400
. 5^
40. f I
68
^?
62
59
55
5*
49
*!
43
'i4'<?
40
4P
17
34
31
28
?S
22
19
16
»S
1420
I 5P ^o
07 04 1
01
98
9%
9«
88
85
8;i
14JO
•H
•
'58 StneesL^jr. I««r. ?t4, 24. »5, a6
.47,28,29, 30, 31. J
84.0
10
so
30
40
50
25.0
10
20
30
40
_!?
26.0
10
20
30
40
27.0
10
20
30
40
28.0
10
20
30
40
-52
29.0
10
20
30
40
30.0
10
20
30
40
-J°
31."?
10
20
30
40
50
39.79
49
38.90
60
—11
03
37.73
!l
36.88
60
32
35-76
49
22
34.95
68
41
33.88
62
36
>IO
32.84
59
33
08
31.83
58
. 33
08
• 83
59
34
10
29.86
62
39
„ »s
28.91
68
45
21
98
75
S3
1
«7
87
11
99
70
42
sz
29
01
"9
9*
65
38
12
86
59
y
07
82
56
3»
P
80
55
30
V,
56
32
08
It
36
12
89
66
42
96
73
50
'73.
43
5:
54
15
96
68
39
10
82
5f
s6
98
71
n
89
*i
36
09
83
57
i»
05
79
53
28
03
z?
S3
28
03
78
S4
30
81
58
34
10
87
63
40
>7
94
7'
48
i.
70
40
II
81
5«
22
36
08
79
5«
»3
96
68
4«
>4
.87
63
33
07
80
y
02.
76
51
25
00
13
so
*5
01
76
5*
27
03
79
51
3>
08
84
61
38
«S
92
4
tf7
11
7t
49
20
9>
4x
33
05
77
1?
21
38
II
84
57
31
04
77
51
*1
00
JS
23
98
-73
48
a3
98
73
49
1*
01
77
S3
29
OS
82
59
35
12
89
68
44
5
34
OS
75
46
«7
88
59
30
02
U
46
fs
90
63
II
81 ■
54
28
01
73
49
13
97
71
4<5
20
9S
70
45
20
96
71
47
9S
74
SO
*7
oj
80
76
33
lo
64
4«
6
61
3>
03
72
43
«4
;i
«7
99
71
43
»5
7?
5*
as
99
46
20
^*
69
43
18
93
68
43
18
93
69
44
20
96
7»
48
«4
01
7?
54
31
08
l^
62
3^J
7
5«
28
99
69
40
II
82
$3
51
68
1?
12
85
S7
30
?i
49
'1
96
70
44
18
9a
66
41
«5
90
61
40
»S
91
66
4*
18
9T
:i
22
98
28
82
60
37
2
55
66
79
$0
22
?!
!Z
10
82
5$
27
00
73
46
20
93
67
41
«S
64
38
'J
88
63
38
i3
88
64-
39
>5
I
43
20
96
73
26
03
80
57
35
9
22
63
34
25
76
47
>9
%
07
79
s*
»S
97
L'
44
«7
9»
^5
3«
•3
11
36
10
60
3?
10
«6
61
37
11
b9
65
4'
17
94
7f
47
01
78
5$
3*
1440
1450
1460
1470
1480
!i9o
1500
1510
1520
1530
1540
1550
1560
1570
1580
1590
lOOO
1610
1^20
163a
1640
1650
1660
1670
1680
1690
170P
1710
1720
1730
1740
1750
1760
1770
1780
1790
1800
1810'
1820
1830
1840
JI8S0
i860
1870
1880
1890
1900
1910
StxtcCsLfgiJ. logar. i-Ay 3^, 33» 34» 35» 3^, 37. 38 ?39- *59 1
1 ft
32.0
10
20
.30
40
50
33.0
10
so
30
40
50
34.0
10
20
30
40
52
35-0
10
20
30
40
—.12
i5.o
10
20
50
40
50
37.0
10
20
30
40
50
38.0
10
20
30
40
50
39.0
10
20
. J®
40
27.30
07
26.8^
63
40
18
25.96
74
J3
3>
10
24.88
67
45
»4
2'
23.82
61
41
20
00
22.79
59
, 39
18
21.98
78
59
39
— 1£
2C.99
80
60
41
32
19.84
] 46
' 27
08
89
i ?5
■ ?J
97
' 79
I
28
P$
83
60
}8
16
94
7>
29
07
86
6$
43
22
01
80
$9
39
18
98
77
57
37
16
96
7«
57
37
17
98
78
59
39
20
01
82
63
44
25
88
69
50
32
»4
95
77
2
25
03
81
58
36
14
92
70 ■
4^
»7
84
52
4»
iO
99
78
57
37
\6
96
75
55
1^
14
94
74
55
35
15
76
$7
\l
9J
80
61
4»
23
04
86
67
49
30
12
94
75
3
01
56
34
12
90
58
4«
*5
11
60
39
97
76
$5
35
14
94
73
53
33
13
9*
72
53
33
13
94
74
55
35
IS
97
78
59
40
21
03
84
65
tl
IO
92
74
_4
21
98
76
54
32
lb
B
66
44
»?
01
80
58
37
id
95
74
53
33
12
91
70
Si
a
10
90
70
51
31
ti
92
7»
53
33
14
95
J9
01
82
45
27
08
9»
7?_
19
96
74
52
29
07
85
44
42
20
99
77
S6
35
14
93
72
51
31
10
89
69
49
29
79
88
69
49
29
09
90
70
5t
32
12
?!
74
11
t8
1?
52
43
25
05
88
70
6
i5
94
72
49
V
05
83
41
18
97
75
54
33
12
91
70
4?
28
08
87
57
47
*7
56
86
67
47
27
07
88
58
49
30
10
91
7*
53
34
t6
97
78
60
41
23
05
85
68
2
14
92
69
47
25
03
81
39
38
J<J
94
li
52
31
10
89
58,
47
26
06
85
tf5
45
25
2*
84
tf5
45
25
05
86
66
47
29
09
89
70
51
33
»4
7i
T»
39
21
03
85
56
8
12
89
67
45
23
01
79
37
35
14
92
u
50
29
08
«7
56
45
24
04
83
63
43
13
02
82
63
43
23
03
f^
54
26
V
87
63
50
3»
12
93
75
5«
38
19
01
83
il
9^
to
87
6$
43
21
99
77
35
33
12
90
69
48
26
05
84
64
43
22
02
81
5i
41
20
00
80
51
41
21
01
82
62
43
24
05
86
67
48
29
10
9»
73
It
99
81
63
1920
1950
1940
'950
.i960
1970
1980
1990
.2000
2010
2020
2030
2040
1050
2060
2070
1080
2090
2100
21 10
2120
2130
■1140
.2i<;o
21 do
21T0
2180
2190
2200
22IO
2220
2230
2240
2250
2260
2270
2280
2290
2300
2310
23*0
2330
2340
2350
2360
2370
2380
2350
Ha
r
N
I
r*5#»i!ffS*~^'Ml^^ 'tS. 44, 45, 46, 47
]t%»\i'
l*^*^*^^*^
m^
\m
£ Sir ^•^i§st#?ii5^=
I
6B
i?
26
13
9$
86
72
1?
46
32
19
06
92
7?
66
53
40
jk2o
3130
?i4o
31 SO
^160
|I70
5180
3190
3200
3210
3220
3230
3240
3250
5260
3270
3z8o
3290
♦6z Street'; Loglji. Ugar. ? xf • 5 &» S7» 58, 59. |
I W
I-
2
3
±
i
6
2
8
9
u
56.0
J. 00
98
97
9^5
t*
93
9«
91
89
88
3360
10
2.9t
8j
84
83
8z
80
7Q
78
76
75
3370
20
74
l^
7»
70
^l
67
66
6j
64
62
3380
50
61
60
5!
>7
56
55
53
5«
s«
50
3390
40
48
47
46
44
43
4»
4'
19
38
37
3400
50
3?
34
?l
f
52
!?
28
27
H
1+
3410
57-
25
Zl
20
«9
18
16
>1
«4
>3
II
34«o
10
10
og
08
06
05
04
02
01
00
99
34JO
ao 1.97
96
95
94
92
9>
go
89
87
96
3440
$0
85
84
82
81
80
79
77
76
75
74
3450
40
P
7'
70
69
67
66
6?
63
62
61
3460
50
60
58
57
5^
55
S±
5J
5'
50
48
34TO
58.0
47
^
45
43
42
4«
40
'2
37
36
3480
»o
35
34
3«
3«
JO
^2
»7
26
H
*4
3490
20
>z
21
20
10
>7
16
•5
14
«a
II
3500
30
10
09
07
06
o.
04
03
01
Oo
29
3510
♦0
99
06
9T
94
93
9«
90
89
88
87
35W
?o
8?
?1
83
<2
80
79
78
77
75
74
3530
59.0
E^
7*
7«
^
68
67
66
64
63
6z
3540
10
6t
(5o
55
57
56
55
53
5*
5<
5°
3550
20
49
47
46
45
44
4«
4»
40
39
38
3560
30
36
35
34
33
31
30
29
28
»7
«5
3570
40
24
*3
22
21
'9
18
17
t6
15
»3
35«o
50
60.0
12
II
to
J
7
6
5
4
2
t
3590
ol
■~\-\
3600
Hours
D««.»
M.
Deg. 'hM.)
Deg.'
I
2 30
I
3
3«
I 18
u
5 oo
2
5
3a
I 20
III
7 3°
3
8
33
I 23
IV
10 oo
4
0^ 10
34
I 25:
V
12 30
5
13
35
I 28
VI
IS oo
6
1,5
36
I 30
VII
>7 3°
■7
18
37
1 33.
VIII
20 oo
8
20
38
1 35'
IX
22 30
9
23
39
I 38
X
25 CO
10
25
40
I , 40
XI
*7 30
II
• 28
4"
I 43
XII
30 00
IS
30
42
I 45
XIII
3« 30
•3
33
43
I 48
XIV
35 00
>4
35
44
I SO
XVi '
37 30
J5
, 38
45
1 53
XVEl
xvil-
40 00
16
c 40
46
i 55
4» 30
»7
43
47
« 58
XVIII
45 0®
18
45
48
2 00
XIX
47 30
•9
48
49
a 3
XX
50 00
20
50
SO
2 5
XXI
s* 30
21
^3
51
2 s>
XXII
55 00
22
55
52
2 10
XXIII
57 30
23
5«
53
2 13
XXIV
60 00
24
I 00
54
2 15
25
« 3
55
* l»
26
I 5
56
2 2C
27
I »
57
2 ^(3
18
1 10
58
2 2^
29
* 13
59
Z 2fe
-
30
1 i$l6o
2 30
/H